Chiral 4-polytopes of type {3,3,k} with automorphism group A_n or S_n for small n ................................................................................. A chiral d-polytope is an abstract polytope of rank d that is maximally symmetric by 'rotation', but admits no reflection; in other words, the automorphism group of the polytope has two orbits on flags of the polytope, such that any two flags which differ in just one element are in different orbits. The automorphism group of a chiral 4-polytope P is generated by three elements x, y and z satisfying the relations x^p = y^q = z^r = (xy)^2 = (yz)^2 = (xyz)^2 = 1, and satisfying an extra property, called the intersection condition. Also P has type {p,q,r}, if p, q and r are the actual orders of x, y and z. By a recent theorem of Marston Conder, Isabel Hubard, Daniel Pellicer and Eugenia O'Reilly-Regueiro, it is known that for all but finitely many integers n > 2, both the alternating group A_n and the symmetric group S_n are isomorphic to the full automorphism group of some chiral 4-polytope of type {3,3,k} for some k, with facets isomorphic to the regular 3-simplex. This theorem and its proof are given in their paper 'Construction of chiral 4-polytopes with alternating or symmetric automorphism group', which has been provisionally accepted for publication in the Journal of Algebraic Combinatorics. The proof in that paper is valid for all n from 50 onwards. It can also be shown that the stated property is true for all n between 20 and 49, and for the alternating group A_n for n = 9, 13, 14, 15, 17 and 18, and for the symmetric group S_n for n = 12, 16, 17, 18 and 19, but for no other values of n. Below is a list showing for each positive integer n up to 49, an example of a generating triple (x,y,z) for A_n and/or S_n as the automorphism group of a chiral 4-polytope of type {3,3,k} for some k, whenever such a polytope exists. Note: the value of k is not unique in most cases; in other words, there can be many such examples for a given degree n. Marston Conder December 2014 ................................................................................. Degree n = 9 A_{9} is the automorphism group of a chiral 4-polytope of type {3,3,9}, via: x = (2, 6, 5)(3, 8, 4), y = (1, 4, 3)(5, 9, 6), z = (1, 5, 8, 6, 7, 9, 3, 2, 4); ................................................................................. Degree n = 12 S_{12} is the automorphism group of a chiral 4-polytope of type {3,3,10}, via: x = (1, 2, 3)(6, 10, 9)(7, 12, 8), y = (2, 3, 5)(4, 8, 7)(9, 11, 10), z = (2, 4, 9, 12, 10, 11, 7, 6, 8, 5); ................................................................................. Degree n = 13 A_{13} is the automorphism group of a chiral 4-polytope of type {3,3,11}, via: x = (1, 2, 3)(6, 10, 9)(7, 12, 8), y = (2, 3, 5)(4, 8, 7)(9, 13, 10), z = (2, 4, 9, 12, 10, 11, 13, 7, 6, 8, 5); ................................................................................. Degree n = 14 A_{14} is the automorphism group of a chiral 4-polytope of type {3,3,35}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 14, 12), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 12, 14), z = (2, 4, 9, 8, 5)(6, 12, 13, 10, 14, 11, 7); ................................................................................. Degree n = 15 A_{15} is the automorphism group of a chiral 4-polytope of type {3,3,9}, via: x = (1, 3, 7)(2, 8, 5)(4, 6, 10)(9, 12, 15)(11, 13, 14), y = (1, 5, 4)(2, 7, 9)(3, 10, 12)(6, 8, 15)(11, 14, 13), z = (1, 4, 2, 11, 15, 8, 10, 7, 5)(3, 14, 9)(6, 13, 12); ................................................................................. Degree n = 16 S_{16} is the automorphism group of a chiral 4-polytope of type {3,3,14}, via: x = (1, 2, 3)(6, 10, 9)(7, 12, 8)(13, 16, 15), y = (2, 3, 5)(4, 8, 7)(9, 14, 10)(11, 13, 16), z = (2, 4, 9, 12, 10, 11, 15, 16, 13, 14, 7, 6, 8, 5); ................................................................................. Degree n = 17 A_{17} is the automorphism group of a chiral 4-polytope of type {3,3,42}, via: x = (1, 2, 3)(6, 13, 10)(7, 15, 9)(8, 14, 16)(11, 17, 12), y = (2, 3, 5)(4, 9, 7)(6, 12, 13)(8, 16, 14)(10, 17, 11), z = (2, 4, 10, 16, 12, 9, 5)(6, 15, 13, 8, 17, 7)(11, 14); S_{17} is the automorphism group of a chiral 4-polytope of type {3,3,28}, via: x = (1, 3, 7)(2, 10, 5)(4, 15, 13)(6, 17, 8)(11, 14, 16), y = (1, 5, 4)(2, 7, 11)(3, 13, 14)(6, 9, 8)(10, 16, 15), z = (1, 8, 12, 9, 13, 7, 5)(2, 14, 6, 4)(3, 11)(10, 17, 15, 16); ................................................................................. Degree n = 18 A_{18} is the automorphism group of a chiral 4-polytope of type {3,3,55}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 17, 16)(12, 14, 18), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 17)(12, 14, 15), z = (2, 4, 9, 8, 5)(6, 12, 15, 16, 13, 10, 18, 17, 14, 11, 7); S_{18} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 5)(3, 9, 8)(4, 12, 6)(7, 16, 11)(10, 17, 14)(13, 18, 15), y = (1, 5, 3)(2, 8, 9)(4, 11, 13)(6, 14, 7)(10, 12, 15)(16, 17, 18), z = (1, 6, 11, 8, 5)(2, 12, 14, 3)(4, 15, 9)(7, 10, 13)(16, 18, 17); ................................................................................. Degree n = 19 S_{19} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 17, 16)(12, 14, 19), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 17)(12, 14, 15), z = (2, 4, 9, 8, 5)(6, 12, 18, 15, 16, 13, 10, 19, 17, 14, 11, 7); ................................................................................. Degree n = 20 A_{20} is the automorphism group of a chiral 4-polytope of type {3,3,12}, via: x = (1, 2, 3)(6, 12, 10)(7, 15, 9)(8, 16, 11)(13, 18, 20)(14, 19, 17), y = (2, 3, 5)(4, 9, 7)(6, 11, 13)(8, 10, 17)(12, 20, 14)(16, 19, 18), z = (2, 4, 10, 11, 9, 5)(6, 13, 8, 14, 18, 19, 16, 20, 15, 12, 17, 7); S_{20} is the automorphism group of a chiral 4-polytope of type {3,3,180}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 16, 17)(12, 14, 19)(15, 20, 18), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 17, 16)(12, 18, 14)(15, 19, 20), z = (2, 4, 9, 8, 5)(6, 12, 17, 13, 10, 19, 18, 11, 7)(14, 15, 20, 16); ................................................................................. Degree n = 21 A_{21} is the automorphism group of a chiral 4-polytope of type {3,3,19}, via: x = (1, 2, 3)(6, 10, 9)(7, 12, 8)(13, 18, 17)(15, 16, 20), y = (2, 3, 5)(4, 8, 7)(9, 14, 10)(11, 15, 16)(17, 21, 18), z = (2, 4, 9, 12, 10, 11, 17, 20, 18, 19, 21, 16, 13, 15, 14, 7, 6, 8, 5); S_{21} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (2, 8, 6)(3, 11, 5)(4, 12, 7)(9, 14, 17)(10, 16, 18)(13, 15, 19), y = (1, 5, 3)(2, 7, 9)(4, 6, 13)(8, 17, 15)(12, 19, 14)(16, 18, 21), z = (1, 6, 7, 5)(2, 9, 4, 10, 13, 3)(8, 18, 19, 12, 17, 11)(14, 16, 20, 21, 15); ................................................................................. Degree n = 22 A_{22} is the automorphism group of a chiral 4-polytope of type {3,3,140}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 21, 22), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 22, 20), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 20, 21, 22, 18, 15); S_{22} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 4, 2)(3, 8, 11)(5, 7, 13)(6, 16, 10)(9, 20, 19)(12, 22, 14)(17, 18, 21), y = (1, 2, 5)(3, 10, 9)(4, 13, 7)(6, 11, 17)(8, 19, 18)(12, 15, 14)(16, 21, 20), z = (1, 3, 14, 15, 19, 13, 4, 11, 10, 5)(6, 18, 12, 9)(7, 8, 17)(16, 22, 20, 21); ................................................................................. Degree n = 23 A_{23} is the automorphism group of a chiral 4-polytope of type {3,3,12}, via: x = (1, 2, 3)(6, 12, 10)(7, 15, 9)(8, 16, 11)(13, 18, 21)(14, 20, 22)(17, 19, 23), y = (2, 3, 5)(4, 9, 7)(6, 11, 13)(8, 10, 17)(12, 21, 19)(14, 22, 20)(16, 23, 18), z = (2, 4, 10, 11, 9, 5)(6, 13, 8, 14, 23, 16, 21, 15, 12, 22, 17, 7)(18, 20, 19); S_{23} is the automorphism group of a chiral 4-polytope of type {3,3,20}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 16, 17)(12, 14, 20)(15, 22, 21)(18, 23, 19), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 17, 16)(12, 19, 14)(15, 21, 22)(18, 20, 23), z = (2, 4, 9, 8, 5)(6, 12, 17, 13, 10, 20, 21, 19, 11, 7)(14, 15, 23, 16)(18, 22); ................................................................................. Degree n = 24 A_{24} is the automorphism group of a chiral 4-polytope of type {3,3,12}, via: x = (1, 2, 3)(6, 12, 10)(7, 15, 9)(8, 16, 11)(13, 18, 21)(14, 20, 22)(17, 19, 23), y = (2, 3, 5)(4, 9, 7)(6, 11, 13)(8, 10, 17)(12, 21, 19)(16, 23, 18)(20, 22, 24), z = (2, 4, 10, 11, 9, 5)(6, 13, 8, 14, 17, 7)(12, 22, 23, 16, 21, 15)(18, 20, 24, 19); S_{24} is the automorphism group of a chiral 4-polytope of type {3,3,210}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 16, 17)(12, 14, 20)(15, 21, 22)(18, 19, 23), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 17, 16)(12, 18, 14)(19, 20, 23)(21, 22, 24), z = (2, 4, 9, 8, 5)(6, 12, 17, 13, 10, 20, 22, 23, 16, 14, 15, 18, 11, 7)(19, 21, 24); ................................................................................. Degree n = 25 A_{25} is the automorphism group of a chiral 4-polytope of type {3,3,140}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 16, 17)(12, 14, 20)(15, 21, 22)(18, 19, 23), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 17, 16)(12, 18, 14)(19, 20, 23)(21, 22, 25), z = (2, 4, 9, 8, 5)(6, 12, 17, 13, 10, 20, 22, 23, 16, 14, 15, 18, 11, 7)(19, 21, 24, 25); S_{25} is the automorphism group of a chiral 4-polytope of type {3,3,30}, via: x = (1, 2, 3)(6, 12, 10)(7, 15, 9)(8, 16, 11)(13, 18, 21)(14, 20, 22)(17, 19, 23), y = (2, 3, 5)(4, 9, 7)(6, 11, 13)(8, 10, 17)(12, 21, 19)(16, 23, 18)(20, 22, 25), z = (2, 4, 10, 11, 9, 5)(6, 13, 8, 14, 17, 7)(12, 22, 23, 16, 21, 15)(18, 20, 24, 25, 19); ................................................................................. Degree n = 26 A_{26} is the automorphism group of a chiral 4-polytope of type {3,3,220}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 23, 24)(21, 25, 22), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 24, 20)(21, 25, 26), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 20, 21, 26, 23, 25, 24, 18, 15); S_{26} is the automorphism group of a chiral 4-polytope of type {3,3,90}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 17, 18)(12, 14, 21)(15, 23, 19)(16, 24, 20)(22, 26, 25), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 18, 17)(12, 20, 15)(14, 19, 22)(16, 21, 25)(23, 24, 26), z = (2, 4, 9, 8, 5)(6, 12, 19, 18, 13, 10, 21, 20, 11, 7)(14, 22, 15, 16, 26, 24, 23, 25, 17); ................................................................................. Degree n = 27 A_{27} is the automorphism group of a chiral 4-polytope of type {3,3,210}, via: x = (1, 2, 3)(6, 12, 10)(7, 15, 9)(8, 17, 11)(13, 21, 23)(14, 22, 24) (16, 19, 25)(18, 20, 26), y = (2, 3, 5)(4, 9, 7)(6, 11, 13)(8, 10, 18)(12, 23, 20)(14, 24, 22) (17, 26, 21)(19, 25, 27), z = (2, 4, 10, 11, 9, 5)(6, 16, 13, 8, 14, 26, 17, 25, 23, 15, 12, 24, 18, 7) (19, 27, 21, 22, 20); S_{27} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 23, 24)(21, 25, 22), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 24, 20)(21, 25, 26), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13) (14, 19, 22, 20, 21, 27, 26, 23, 25, 24, 18, 15); ................................................................................. Degree n = 28 A_{28} is the automorphism group of a chiral 4-polytope of type {3,3,70}, via: x = (1, 2, 3)(6, 10, 9)(7, 12, 8)(13, 20, 19)(15, 22, 18)(16, 17, 24) (21, 25, 27)(23, 26, 28), y = (2, 3, 5)(4, 8, 7)(9, 14, 10)(11, 16, 17)(13, 18, 21)(15, 19, 23) (20, 27, 26)(22, 28, 25), z = (2, 4, 9, 12, 10, 11, 19, 18, 16, 14, 7, 6, 8, 5)(13, 21, 15, 23, 17) (20, 28, 22, 27, 24)(25, 26); S_{28} is the automorphism group of a chiral 4-polytope of type {3,3,420}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 28, 27), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(24, 26, 28), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 28, 25, 26, 27); ................................................................................. Degree n = 29 A_{29} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 16, 17)(12, 14, 20)(15, 22, 21) (18, 25, 19)(23, 27, 28)(24, 26, 29), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 17, 16)(12, 19, 14)(15, 21, 23) (18, 20, 25)(22, 28, 27)(24, 29, 26), z = (2, 4, 9, 8, 5)(6, 12, 17, 13, 10, 20, 21, 19, 11, 7) (14, 15, 24, 28, 25, 16)(18, 22, 29, 23)(26, 27); S_{29} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 17, 18)(12, 14, 21)(15, 24, 19) (16, 20, 25)(22, 27, 29)(23, 26, 28), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 18, 17)(12, 16, 15)(14, 19, 22) (20, 21, 29)(23, 28, 26)(24, 25, 27), z = (2, 4, 9, 8, 5)(6, 12, 19, 18, 13, 10, 21, 25, 24, 28, 22, 15, 16, 11, 7) (14, 23, 29, 17)(20, 26, 27); ................................................................................. Degree n = 30 A_{30} is the automorphism group of a chiral 4-polytope of type {3,3,260}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 23, 24) (21, 25, 22)(28, 30, 29), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 24, 20) (21, 25, 26)(27, 29, 28), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13) (14, 19, 22, 20, 21, 27, 29, 26, 23, 25, 24, 18, 15); S_{30} is the automorphism group of a chiral 4-polytope of type {3,3,130}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 17, 18)(12, 14, 21)(15, 24, 19) (16, 25, 20)(22, 28, 26)(23, 27, 29), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 18, 17)(12, 20, 15)(14, 19, 22)(16, 21, 26) (24, 25, 28)(27, 29, 30), z = (2, 4, 9, 8, 5)(6, 12, 19, 18, 13, 10, 21, 20, 11, 7) (14, 23, 22, 15, 16, 27, 30, 28, 25, 24, 29, 26, 17); ................................................................................. Degree n = 31 A_{31} is the automorphism group of a chiral 4-polytope of type {3,3,70}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 17, 18)(12, 14, 21)(15, 24, 19) (16, 25, 20)(22, 28, 26)(23, 27, 29), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 18, 17)(12, 20, 15)(14, 19, 22) (16, 21, 26)(24, 25, 28)(27, 29, 30), z = (2, 4, 9, 8, 5)(6, 12, 19, 18, 13, 10, 21, 20, 11, 7) (14, 23, 22, 15, 16, 27, 31, 30, 28, 25, 24, 29, 26, 17); S_{31} is the automorphism group of a chiral 4-polytope of type {3,3,120}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 26, 25) (21, 29, 22)(23, 30, 28)(24, 31, 27), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 25, 20) (21, 28, 30)(22, 29, 23)(24, 27, 31), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13) (14, 19, 22, 27, 28, 25, 18, 15)(20, 21, 31, 23, 26, 29)(24, 30); ................................................................................. Degree n = 32 A_{32} is the automorphism group of a chiral 4-polytope of type {3,3,660}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 16, 17)(12, 14, 21)(15, 23, 25) (18, 26, 22)(19, 20, 29)(24, 28, 31)(27, 30, 32), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 17, 16)(12, 19, 14)(15, 22, 24) (18, 25, 27)(20, 21, 29)(23, 31, 30)(26, 32, 28), z = (2, 4, 9, 8, 5)(6, 12, 17, 13, 10, 21, 25, 22, 19, 11, 7) (14, 15, 24, 18, 27, 20, 23, 32, 26, 31, 29, 16)(28, 30); S_{32} is the automorphism group of a chiral 4-polytope of type {3,3,660}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 30, 31)(27, 29, 32), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(24, 28, 31)(26, 32, 29), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 29, 30, 32, 25, 26, 31, 28, 27); ................................................................................. Degree n = 33 A_{33} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 30, 32)(27, 29, 28), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(24, 30, 31)(26, 28, 29), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 33, 31, 28, 25, 26, 32, 29, 30, 27); S_{33} is the automorphism group of a chiral 4-polytope of type {3,3,340}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 17, 18)(12, 14, 22)(15, 25, 19) (16, 26, 27)(20, 21, 30)(23, 29, 32)(24, 28, 33), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 18, 17)(12, 20, 15)(14, 19, 23) (21, 22, 32)(24, 33, 28)(25, 30, 29)(26, 27, 31), z = (2, 4, 9, 8, 5)(6, 12, 19, 18, 13, 10, 22, 27, 30, 25, 33, 23, 15, 16, 20, 11, 7) (14, 24, 32, 17)(21, 28, 29, 26, 31); ................................................................................. Degree n = 34 A_{34} is the automorphism group of a chiral 4-polytope of type {3,3,180}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 29, 31)(27, 28, 32)(30, 34, 33), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(24, 31, 30)(26, 32, 28)(29, 33, 34), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 33, 32, 25, 26, 31, 27)(28, 29, 34, 30); S_{34} is the automorphism group of a chiral 4-polytope of type {3,3,420}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 26, 24) (21, 28, 22)(23, 31, 27)(25, 33, 30)(29, 32, 34), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 24, 20) (21, 27, 29)(22, 30, 23)(25, 28, 34)(31, 33, 32), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 27, 24, 18, 15) (20, 21, 29, 23, 25, 32, 33, 31, 34, 26, 28, 30); ................................................................................. Degree n = 35 A_{35} is the automorphism group of a chiral 4-polytope of type {3,3,660}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 16, 17)(12, 14, 21)(15, 23, 25) (18, 26, 22)(19, 20, 29)(24, 28, 32)(27, 33, 34)(30, 31, 35), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 17, 16)(12, 19, 14)(15, 22, 24) (18, 25, 27)(20, 21, 29)(23, 32, 33)(26, 34, 28)(30, 35, 31), z = (2, 4, 9, 8, 5)(6, 12, 17, 13, 10, 21, 25, 22, 19, 11, 7) (14, 15, 24, 18, 30, 34, 26, 32, 29, 16)(20, 23, 35, 27)(28, 31, 33); S_{35} is the automorphism group of a chiral 4-polytope of type {3,3,20}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 23, 24) (21, 25, 22)(28, 31, 32)(29, 33, 30), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 24, 20) (21, 25, 26)(27, 32, 28)(29, 33, 34), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13) (14, 19, 22, 20, 21, 27, 30, 28, 29, 35, 34, 31, 33, 32, 26, 23, 25, 24, 18, 15); ................................................................................. Degree n = 36 A_{36} is the automorphism group of a chiral 4-polytope of type {3,3,1540}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 16, 17)(12, 14, 21)(15, 23, 25) (18, 26, 22)(19, 20, 29)(24, 28, 32)(27, 33, 34)(30, 31, 35), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 17, 16)(12, 19, 14)(15, 22, 24) (18, 25, 27)(20, 21, 29)(23, 32, 33)(26, 34, 28)(31, 35, 36), z = (2, 4, 9, 8, 5)(6, 12, 17, 13, 10, 21, 25, 22, 19, 11, 7) (14, 15, 24, 18, 30, 27, 20, 23, 35, 34, 26, 32, 29, 16)(28, 31, 36, 33); S_{36} is the automorphism group of a chiral 4-polytope of type {3,3,420}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 29, 30)(27, 28, 31)(34, 35, 36), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(26, 31, 28)(29, 30, 33)(32, 34, 35), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 27)(25, 26, 30, 31)(28, 29, 32, 36, 35, 34, 33); ................................................................................. Degree n = 37 A_{37} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 29, 31)(27, 28, 33)(30, 36, 35)(32, 37, 34), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(24, 31, 30)(26, 33, 28)(29, 35, 36)(32, 34, 37), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 34, 35, 33, 25, 26, 31, 27)(28, 29, 37, 30)(32, 36); S_{37} is the automorphism group of a chiral 4-polytope of type {3,3,280}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 26, 24) (21, 28, 22)(23, 32, 27)(25, 34, 30)(29, 36, 35)(31, 33, 37), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 24, 20) (21, 27, 29)(22, 30, 23)(25, 28, 35)(31, 37, 33)(32, 34, 36), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 27, 24, 18, 15) (20, 21, 31, 35, 26, 28, 30)(23, 25, 33, 36, 34, 32, 37, 29); ................................................................................. Degree n = 38 A_{38} is the automorphism group of a chiral 4-polytope of type {3,3,420}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 29, 31)(27, 28, 33)(30, 35, 36)(32, 37, 34), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(24, 31, 30)(26, 33, 28)(29, 36, 35)(32, 37, 38), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 34, 30, 28, 29, 37, 36, 33, 25, 26, 31, 27)(32, 38, 35); S_{38} is the automorphism group of a chiral 4-polytope of type {3,3,560}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 26, 24) (21, 28, 22)(23, 32, 27)(25, 34, 30)(29, 36, 35)(31, 33, 37), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 24, 20) (21, 27, 29)(22, 30, 23)(25, 28, 35)(32, 34, 36)(33, 37, 38), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 27, 24, 18, 15) (20, 21, 31, 29, 23, 25, 33, 38, 36, 34, 32, 37, 35, 26, 28, 30); ................................................................................. Degree n = 39 A_{39} is the automorphism group of a chiral 4-polytope of type {3,3,2380}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 26, 24) (21, 28, 22)(23, 32, 27)(25, 34, 30)(29, 36, 35)(31, 33, 37), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 24, 20) (21, 27, 29)(22, 30, 23)(25, 28, 35)(32, 34, 36)(33, 37, 38), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 27, 24, 18, 15) (20, 21, 31, 29, 23, 25, 33, 39, 38, 36, 34, 32, 37, 35, 26, 28, 30); S_{39} is the automorphism group of a chiral 4-polytope of type {3,3,420}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 29, 31)(27, 28, 33)(30, 35, 36)(32, 37, 34), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(24, 31, 30)(26, 33, 28)(29, 36, 35)(32, 37, 38), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 34, 30, 28, 29, 37, 36, 33, 25, 26, 31, 27)(32, 39, 38, 35); ................................................................................. Degree n = 40 A_{40} is the automorphism group of a chiral 4-polytope of type {3,3,180}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 30, 33)(27, 29, 34)(28, 36, 32)(31, 40, 35)(37, 38, 39), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(24, 32, 31)(26, 34, 29)(28, 33, 37)(30, 35, 38)(36, 39, 40), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 35, 34, 25, 26, 33, 32, 27)(28, 37, 29, 30, 39, 36, 40, 38, 31); S_{40} is the automorphism group of a chiral 4-polytope of type {3,3,660}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 29, 30)(27, 28, 31)(34, 38, 37)(35, 40, 36), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(26, 31, 28)(29, 30, 33)(32, 36, 35)(37, 39, 38), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 27)(25, 26, 30, 31)(28, 29, 32, 37, 40, 38, 39, 35, 34, 36, 33); ................................................................................. Degree n = 41 A_{41} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 29, 30)(27, 28, 31)(34, 38, 37)(35, 40, 36), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(26, 31, 28)(29, 30, 33)(32, 36, 35)(37, 41, 38), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 27)(25, 26, 30, 31)(28, 29, 32, 37, 40, 38, 39, 41, 35, 34, 36, 33); S_{41} is the automorphism group of a chiral 4-polytope of type {3,3,140}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 26, 24) (21, 28, 22)(23, 33, 27)(25, 35, 36)(29, 37, 38)(30, 31, 39)(32, 34, 41), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 24, 20) (21, 27, 29)(22, 30, 23)(28, 38, 31)(32, 41, 34)(33, 39, 37)(35, 36, 40), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 27, 24, 18, 15) (20, 21, 32, 38, 26, 28, 36, 39, 33, 41, 29, 23, 25, 30)(31, 34, 37, 35, 40); ................................................................................. Degree n = 42 A_{42} is the automorphism group of a chiral 4-polytope of type {3,3,420}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23) (21, 26, 22)(24, 29, 30)(27, 28, 31)(34, 38, 37)(35, 41, 36)(39, 42, 40), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20) (21, 22, 27)(26, 31, 28)(29, 30, 33)(32, 36, 35)(34, 37, 39)(38, 40, 42), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 27)(25, 26, 30, 31)(28, 29, 32, 37, 36, 33)(34, 40, 41, 38, 42, 39, 35); S_{42} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 23, 24) (21, 25, 22)(28, 34, 32)(29, 36, 30)(31, 39, 35)(33, 41, 38)(37, 40, 42), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 24, 20) (21, 25, 26)(27, 32, 28)(29, 35, 37)(30, 38, 31)(33, 36, 42)(39, 41, 40), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13) (14, 19, 22, 20, 21, 27, 30, 35, 32, 26, 23, 25, 24, 18, 15) (28, 29, 37, 31, 33, 40, 41, 39, 42, 34, 36, 38); ................................................................................. Degree n = 43 A_{43} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 30, 33)(27, 29, 35)(28, 36, 32)(31, 34, 39)(37, 40, 43)(38, 41, 42), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (24, 32, 31)(26, 35, 29)(28, 33, 37)(30, 39, 40)(34, 36, 43)(38, 42, 41), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 31, 28, 38, 43, 36, 39, 35, 25, 26, 33, 32, 27)(29, 30, 42, 37) (34, 41, 40); S_{43} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 29, 31)(27, 28, 33)(30, 36, 35)(32, 38, 34)(37, 41, 42)(39, 40, 43), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (24, 31, 30)(26, 33, 28)(29, 35, 36)(32, 34, 39)(37, 42, 41)(38, 43, 40), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 34, 35, 33, 25, 26, 31, 27)(28, 29, 38, 42, 39, 30) (32, 37, 43, 36)(40, 41); ................................................................................. Degree n = 44 A_{44} is the automorphism group of a chiral 4-polytope of type {3,3,780}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 30, 33)(27, 29, 34)(28, 36, 32)(31, 41, 35)(37, 39, 44)(38, 42, 40), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (24, 32, 31)(26, 34, 29)(28, 33, 37)(30, 35, 39)(36, 44, 41)(40, 43, 42), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 35, 34, 25, 26, 33, 32, 27) (28, 38, 37, 29, 30, 40, 44, 36, 41, 42, 43, 39, 31); S_{44} is the automorphism group of a chiral 4-polytope of type {3,3,780}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 29, 30)(27, 28, 31)(34, 38, 37)(35, 40, 36)(41, 44, 43), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (26, 31, 28)(29, 30, 33)(32, 36, 35)(37, 42, 38)(39, 43, 41), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 27)(25, 26, 30, 31)(28, 29, 32, 37, 40, 38, 39, 43, 42, 35, 34, 36, 33); ................................................................................. Degree n = 45 A_{45} is the automorphism group of a chiral 4-polytope of type {3,3,120}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 29, 30)(27, 28, 31)(34, 41, 38)(35, 43, 37)(36, 42, 44)(39, 45, 40), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (26, 31, 28)(29, 30, 33)(32, 37, 35)(34, 40, 41)(36, 44, 42)(38, 45, 39), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 27)(25, 26, 30, 31)(28, 29, 32, 38, 44, 40, 37, 33) (34, 43, 41, 36, 45, 35)(39, 42); S_{45} is the automorphism group of a chiral 4-polytope of type {3,3,420}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 30, 33)(27, 29, 34)(28, 36, 32)(31, 41, 35)(37, 39, 44)(38, 42, 40), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (24, 32, 31)(26, 34, 29)(28, 33, 37)(30, 35, 39)(36, 44, 41)(40, 45, 42), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 35, 34, 25, 26, 33, 32, 27) (28, 38, 37, 29, 30, 40, 44, 36, 41, 42, 43, 45, 39, 31); ................................................................................. Degree n = 46 A_{46} is the automorphism group of a chiral 4-polytope of type {3,3,660}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 29, 30)(27, 28, 31)(34, 38, 37)(35, 41, 36)(39, 45, 44)(40, 42, 46) y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (26, 31, 28)(29, 30, 33)(32, 36, 35)(34, 37, 39)(38, 44, 45)(40, 42, 43) z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 27)(25, 26, 30, 31)(28, 29, 32, 37, 36, 33) (34, 40, 43, 44, 41, 38, 46, 45, 42, 39, 35) S_{46} is the automorphism group of a chiral 4-polytope of type {3,3,1980}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 29, 31)(27, 28, 33)(30, 37, 36)(32, 40, 34)(35, 43, 39)(38, 45, 42)(41, 44, 46) y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (24, 31, 30)(26, 33, 28)(29, 36, 37)(32, 39, 41)(34, 42, 35)(38, 40, 46)(43, 45, 44) z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 34, 39, 36, 33, 25, 26, 31, 27)(28, 29, 40, 42, 30) (32, 41, 35, 38, 44, 45, 43, 46, 37) ................................................................................. Degree n = 47 A_{47} is the automorphism group of a chiral 4-polytope of type {3,3,660}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 30, 33)(27, 29, 35)(28, 37, 32)(31, 42, 40)(34, 45, 36)(38, 41, 46)(39, 43, 44) y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (24, 32, 31)(26, 35, 29)(28, 33, 38)(30, 40, 41)(34, 36, 45)(37, 46, 42)(43, 44, 47) z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 36, 40, 35, 25, 26, 33, 32, 27)(28, 39, 38, 29, 30, 44, 46, 37, 45, 31) (34, 42, 43, 47, 41) S_{47} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 29, 30)(27, 28, 31)(34, 38, 37)(35, 41, 36)(39, 45, 44)(40, 42, 47) y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (26, 31, 28)(29, 30, 33)(32, 36, 35)(34, 37, 39)(38, 44, 45)(40, 42, 43) z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15) (20, 21, 24, 27)(25, 26, 30, 31)(28, 29, 32, 37, 36, 33) (34, 40, 46, 43, 44, 41, 38, 47, 45, 42, 39, 35) ................................................................................. Degree n = 48 A_{48} is the automorphism group of a chiral 4-polytope of type {3,3,420}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 29, 30)(27, 28, 31)(34, 40, 38)(35, 43, 37)(36, 44, 39)(41, 46, 48)(42, 47, 45), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (26, 31, 28)(29, 30, 33)(32, 37, 35)(34, 39, 41)(36, 38, 45)(40, 48, 42)(44, 47, 46), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15)(20, 21, 24, 27) (25, 26, 30, 31)(28, 29, 32, 38, 39, 37, 33)(34, 41, 36, 42, 46, 47, 44, 48, 43, 40, 45, 35); S_{48} is the automorphism group of a chiral 4-polytope of type {3,3,180}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 29, 30)(27, 28, 31)(34, 38, 37)(35, 41, 36)(39, 44, 45)(40, 42, 47)(43, 48, 46), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (26, 31, 28)(29, 30, 33)(32, 36, 35)(34, 37, 39)(38, 45, 44)(40, 46, 42)(43, 47, 48), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15)(20, 21, 24, 27) (25, 26, 30, 31)(28, 29, 32, 37, 36, 33)(34, 40, 45, 41, 38, 47, 46, 39, 35)(42, 43, 48, 44); ................................................................................. Degree n = 49 A_{49} is the automorphism group of a chiral 4-polytope of type {3,3,60}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 29, 30)(27, 28, 31)(34, 38, 37)(35, 40, 36)(41, 46, 45)(43, 44, 48), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (26, 31, 28)(29, 30, 33)(32, 36, 35)(37, 42, 38)(39, 43, 44)(45, 49, 46), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15)(20, 21, 24, 27) (25, 26, 30, 31)(28, 29, 32, 37, 40, 38, 39, 45, 48, 46, 47, 49, 44, 41, 43, 42, 35, 34, 36, 33); S_{49} is the automorphism group of a chiral 4-polytope of type {3,3,1320}, via: x = (1, 2, 3)(6, 10, 9)(7, 13, 8)(11, 15, 16)(12, 14, 17)(20, 25, 23)(21, 26, 22) (24, 29, 31)(27, 28, 33)(30, 37, 36)(32, 40, 34)(35, 44, 39)(38, 46, 42)(41, 48, 47)(43, 45, 49), y = (2, 3, 5)(4, 8, 7)(6, 9, 11)(10, 16, 15)(14, 17, 18)(19, 23, 20)(21, 22, 27) (24, 31, 30)(26, 33, 28)(29, 36, 37)(32, 39, 41)(34, 42, 35)(38, 40, 47)(43, 49, 45)(44, 46, 48), z = (2, 4, 9, 8, 5)(6, 12, 11, 7)(10, 17, 16, 13)(14, 19, 22, 23, 18, 15)(20, 21, 24, 34, 39, 36, 33, 25, 26, 31, 27)(28, 29, 40, 42, 30)(32, 43, 47, 37)(35, 38, 45, 48, 46, 44, 49, 41). ....................................................................................