Magma V2.19-9 Wed Oct 14 2015 12:15:21 on trit2 [Seed = 3220674084] Type ? for help. Type -D to quit. > g1:=Alt(8); > g1; Permutation group g1 acting on a set of cardinality 8 Order = 20160 = 2^6 * 3^2 * 5 * 7 (1, 2)(3, 4, 5, 6, 7, 8) (1, 2, 3) > Classes(g1); Conjugacy Classes of group g1 ----------------------------- [1] Order 1 Length 1 Rep Id(g1) [2] Order 2 Length 105 Rep (1, 2)(3, 4)(5, 6)(7, 8) [3] Order 2 Length 210 Rep (1, 2)(3, 4) [4] Order 3 Length 112 Rep (1, 2, 3) [5] Order 3 Length 1120 Rep (1, 2, 3)(4, 5, 6) [6] Order 4 Length 1260 Rep (1, 2, 3, 4)(5, 6, 7, 8) [7] Order 4 Length 2520 Rep (1, 2, 3, 4)(5, 6) [8] Order 5 Length 1344 Rep (1, 2, 3, 4, 5) [9] Order 6 Length 1680 Rep (1, 2, 3)(4, 5)(6, 7) [10] Order 6 Length 3360 Rep (1, 2, 3, 4, 5, 6)(7, 8) [11] Order 7 Length 2880 Rep (1, 2, 3, 4, 5, 6, 7) [12] Order 7 Length 2880 Rep (1, 3, 4, 5, 6, 7, 2) [13] Order 15 Length 1344 Rep (1, 2, 3, 4, 5)(6, 7, 8) [14] Order 15 Length 1344 Rep (1, 3, 4, 5, 2)(6, 7, 8) > G2:=GL(4,2); > G2; GL(4, GF(2)) > Classes(G2); Conjugacy Classes of group G2 ----------------------------- [1] Order 1 Length 1 Rep [1 0 0 0] [0 1 0 0] [0 0 1 0] =I4, identity [0 0 0 1] [2] Order 2 Length 105 Rep [1 0 0 0] [0 1 0 0] [0 0 1 1] =(x^2 - 1) [0 0 0 1] [3] Order 2 Length 210 Rep [1 1 0 0] [0 1 0 0] =(x^2 - 1),(x^2 - 1) [0 0 1 1] [0 0 0 1] [4] Order 3 Length 112 Rep [0 1 0 0] [1 1 0 0] =(x^2+x+1),(x^2+x+1) [0 0 0 1] [0 0 1 1] [5] Order 3 Length 1120 Rep [1 0 0 0] [0 1 0 0] =(x^2+x+1) [0 0 0 1] [0 0 1 1] [6] Order 4 Length 1260 Rep [1 0 0 0] [0 1 1 0] [0 0 1 1] [0 0 0 1] [7] Order 4 Length 2520 Rep [1 1 0 0] [0 1 1 0] [0 0 1 1] [0 0 0 1] [8] Order 5 Length 1344 Rep [0 1 0 0] [0 0 1 0] [0 0 0 1] =x^4+x^3+x^2+x+1 [1 1 1 1] [9] Order 6 Length 1680 Rep [0 1 0 0] [1 1 1 0] [0 0 0 1] [0 0 1 1] [10] Order 6 Length 3360 Rep [1 1 0 0] [0 1 0 0] [0 0 0 1] [0 0 1 1] [11] Order 7 Length 2880 Rep [1 0 0 0] [0 0 1 0] [0 0 0 1] [0 1 1 0] [12] Order 7 Length 2880 Rep [1 0 0 0] [0 0 1 0] [0 0 0 1] [0 1 0 1] [13] Order 15 Length 1344 Rep [0 1 0 0] [0 0 1 0] [0 0 0 1] [1 1 0 0] [14] Order 15 Length 1344 Rep [0 1 0 0] [0 0 1 0] [0 0 0 1] [1 0 0 1] > G3:=PSL(3,4); > G3; Permutation group G3 acting on a set of cardinality 21 Order = 20160 = 2^6 * 3^2 * 5 * 7 (4, 11, 20)(5, 15, 17)(6, 16, 18)(7, 14, 13)(8, 12, 9)(10, 21, 19) (1, 8, 21, 16, 15, 3, 2)(4, 10, 20, 18, 17, 9, 7)(5, 12, 11, 14, 19, 13, 6) > Classes(G3); Conjugacy Classes of group G3 ----------------------------- [1] Order 1 Length 1 Rep Id(G3) [2] Order 2 Length 315 Rep (2, 17)(3, 8)(4, 21)(5, 15)(9, 12)(11, 19)(13, 18)(14, 16) [3] Order 3 Length 2240 Rep (1, 18, 19)(2, 21, 8)(3, 6, 9)(4, 20, 14)(5, 12, 16)(10, 15, 17) [4] Order 4 Length 1260 Rep (2, 9, 17, 12)(3, 15, 8, 5)(4, 11, 21, 19)(6, 7)(10, 20)(13, 16, 18, 14) [5] Order 4 Length 1260 Rep (2, 13, 17, 18)(3, 4, 8, 21)(5, 11, 15, 19)(6, 10)(7, 20)(9, 14, 12, 16) [6] Order 4 Length 1260 Rep (2, 21, 17, 4)(3, 13, 8, 18)(5, 16, 15, 14)(6, 20)(7, 10)(9, 11, 12, 19) [7] Order 5 Length 4032 Rep (1, 5, 14, 18, 6)(2, 10, 15, 3, 7)(4, 8, 13, 19, 21)(9, 16, 20, 17, 11) [8] Order 5 Length 4032 Rep (1, 14, 6, 5, 18)(2, 15, 7, 10, 3)(4, 13, 21, 8, 19)(9, 20, 11, 16, 17) [9] Order 7 Length 2880 Rep (1, 15, 19, 2, 8, 12, 11)(3, 4, 13, 17, 10, 14, 21)(5, 16, 20, 7, 9, 6, 18) [10] Order 7 Length 2880 Rep (1, 2, 11, 19, 12, 15, 8)(3, 17, 21, 13, 14, 4, 10)(5, 7, 18, 20, 6, 16, 9) ------------ [1] Order 1 Length 1 [2] Order 2 Length 105 [3] Order 2 Length 210 [4] Order 3 Length 112 [5] Order 3 Length 1120 [6] Order 4 Length 1260 [7] Order 4 Length 2520 [8] Order 5 Length 1344 [9] Order 6 Length 1680 [10] Order 6 Length 3360 [11] Order 7 Length 2880 [12] Order 7 Length 2880 [13] Order 15 Length 1344 [14] Order 15 Length 1344 1+105+210+112+1120+1260+2520+1344+1680+3360+2880+2880+1344+1344 = 20160 [1] Order 1 Length 1 [2] Order 2 Length 315 [3] Order 3 Length 2240 [4] Order 4 Length 1260 [5] Order 4 Length 1260 [6] Order 4 Length 1260 [7] Order 5 Length 4032 [8] Order 5 Length 4032 [9] Order 7 Length 2880 [10] Order 7 Length 2880 1+315+2240+1260+1260+1260+4032+4032+2880+2880 = 20160 G3 has no order six or fifteen elements! ------------ Compare 2-Sylow subgroups: > GP2:=Sylow(G1,2); > Classes(GP2); Conjugacy Classes of group GP2 [16 classes] ------------------------------ 1+1+2+2+2+4+4+4+4+4+4+4+4+8+8+8 = 64, Center = Z2 > GP3:=Sylow(G3,2); > Classes(GP3); Conjugacy Classes of group GP3 [19 classes] ------------------------------ 1+1+1+1+4+4+4+4+4+4+4+4+4+4+4+4+4+4+4 = 64, Center = Z2xZ2 Nonisomorphic 2-Sylow centers (so nonisom 2-Sylows)