Magma V2.20-3 Fri Mar 18 2016 10:20:24 on Fordham-David-Swinarski [Seed = 2569257324] Type ? for help. Type -D to quit. > G:=SmallGroup(48,14); > load "autcv10c.txt"; Loading "autcv10c.txt" > RunExample(G,5,[2,4,12]); Set seed to 0. Character Table of Group G -------------------------- ------------------------------------------------------------------------------ Class | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Size | 1 1 1 1 6 6 2 2 2 6 6 2 2 2 2 2 2 2 Order | 1 2 2 2 2 2 3 4 4 4 4 6 6 6 12 12 12 12 ------------------------------------------------------------------------------ p = 2 1 1 1 1 1 1 7 3 3 2 2 7 7 7 14 14 14 14 p = 3 1 2 3 4 5 6 1 9 8 11 10 4 2 3 8 8 9 9 ------------------------------------------------------------------------------ X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 + 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 X.3 + 1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 X.4 + 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 X.5 0 1 -1 -1 1 1 -1 1 I -I -I I 1 -1 -1 -I -I I I X.6 0 1 -1 -1 1 -1 1 1 I -I I -I 1 -1 -1 -I -I I I X.7 0 1 -1 -1 1 -1 1 1 -I I -I I 1 -1 -1 I I -I -I X.8 0 1 -1 -1 1 1 -1 1 -I I I -I 1 -1 -1 I I -I -I X.9 + 2 2 2 2 0 0 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 X.10 + 2 2 2 2 0 0 -1 -2 -2 0 0 -1 -1 -1 1 1 1 1 X.11 + 2 2 -2 -2 0 0 2 0 0 0 0 -2 2 -2 0 0 0 0 X.12 + 2 -2 2 -2 0 0 2 0 0 0 0 -2 -2 2 0 0 0 0 X.13 0 2 -2 2 -2 0 0 -1 0 0 0 0 1 1 -1-1-2*J 1+2*J-1-2*J 1+2*J X.14 0 2 -2 2 -2 0 0 -1 0 0 0 0 1 1 -1 1+2*J-1-2*J 1+2*J-1-2*J X.15 0 2 -2 -2 2 0 0 -1 -2*I 2*I 0 0 -1 1 1 -I -I I I X.16 0 2 -2 -2 2 0 0 -1 2*I -2*I 0 0 -1 1 1 I I -I -I X.17 + 2 2 -2 -2 0 0 -1 0 0 0 0 1 -1 1 Z1 -Z1 -Z1 Z1 X.18 + 2 2 -2 -2 0 0 -1 0 0 0 0 1 -1 1 -Z1 Z1 Z1 -Z1 Explanation of Character Value Symbols -------------------------------------- # denotes algebraic conjugation, that is, #k indicates replacing the root of unity w by w^k J = RootOfUnity(3) I = RootOfUnity(4) Z1 = (CyclotomicField(12: Sparse := true)) ! [ RationalField() | 0, 1, 0, 2 ] Conjugacy Classes of group G ---------------------------- [1] Order 1 Length 1 Rep Id(G) [2] Order 2 Length 1 Rep G.3 * G.4 [3] Order 2 Length 1 Rep G.3 [4] Order 2 Length 1 Rep G.4 [5] Order 2 Length 6 Rep G.1 [6] Order 2 Length 6 Rep G.1 * G.3 [7] Order 3 Length 2 Rep G.5 [8] Order 4 Length 2 Rep G.2 [9] Order 4 Length 2 Rep G.2 * G.3 [10] Order 4 Length 6 Rep G.1 * G.2 * G.3 * G.4 [11] Order 4 Length 6 Rep G.1 * G.2 * G.4 [12] Order 6 Length 2 Rep G.4 * G.5 [13] Order 6 Length 2 Rep G.3 * G.4 * G.5 [14] Order 6 Length 2 Rep G.3 * G.5 [15] Order 12 Length 2 Rep G.2 * G.3 * G.5 [16] Order 12 Length 2 Rep G.2 * G.3 * G.5^2 [17] Order 12 Length 2 Rep G.2 * G.5 [18] Order 12 Length 2 Rep G.2 * G.5^2 Surface kernel generators: [ G.1 * G.4 * G.5^2, G.1 * G.2 * G.4, G.2 * G.3 * G.5^2 ] Is hyperelliptic? true Curve is hyperelliptic > FP,f:=FPGroup(G); > PermG,g:=PermutationGroup(FP); > g(Inverse(f)(G.1 * G.4 * G.5^2)); (1, 20)(2, 17)(3, 24)(4, 12)(5, 11)(6, 16)(7, 23)(9, 19)(10, 18)(13, 22) > g(Inverse(f)(G.1 * G.2 * G.4)); (1, 6, 9, 7)(2, 3, 13, 4)(5, 16, 18, 23)(8, 19, 21, 20)(10, 22, 11, 17)(12, 14, 24, 15) > PermG; Permutation group PermG acting on a set of cardinality 24 Order = 48 = 2^4 * 3 (2, 6)(5, 12)(7, 13)(8, 16)(10, 19)(11, 20)(14, 17)(15, 22)(18, 24)(21, 23) (1, 2, 3, 7)(4, 6, 9, 13)(5, 8, 10, 15)(11, 14, 18, 21)(12, 17, 19, 23)(16, 24, 22, 20) (1, 3)(2, 7)(4, 9)(5, 10)(6, 13)(8, 15)(11, 18)(12, 19)(14, 21)(16, 22)(17, 23)(20, 24) (1, 4)(2, 6)(3, 9)(5, 11)(7, 13)(8, 14)(10, 18)(12, 20)(15, 21)(16, 17)(19, 24)(22, 23) (1, 5, 12)(2, 8, 17)(3, 10, 19)(4, 11, 20)(6, 14, 16)(7, 15, 23)(9, 18, 24)(13, 21, 22)