Magma V2.20-3 Fri Mar 18 2016 11:07:10 on Fordham-David-Swinarski [Seed = 2937446507] Type ? for help. Type -D to quit. > load "autcv10c.txt"; Loading "autcv10c.txt" > G:=SmallGroup(40,5); > RunExample(G,5,[2,4,20]); 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 Size | 1 1 5 5 1 1 5 5 2 2 2 2 2 2 2 2 Order | 1 2 2 2 4 4 4 4 5 5 10 10 20 20 20 20 ---------------------------------------------------------------------------- p = 2 1 1 1 1 2 2 2 2 10 9 9 10 11 11 12 12 p = 5 1 2 3 4 5 6 7 8 1 1 2 2 5 6 6 5 ---------------------------------------------------------------------------- X.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 X.3 + 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 X.5 0 1 -1 -1 1 I -I -I I 1 1 -1 -1 I -I -I I X.6 0 1 -1 1 -1 I -I I -I 1 1 -1 -1 I -I -I I X.7 0 1 -1 1 -1 -I I -I I 1 1 -1 -1 -I I I -I X.8 0 1 -1 -1 1 -I I I -I 1 1 -1 -1 -I I I -I X.9 + 2 2 0 0 2 2 0 0 Z1 Z1#2 Z1#2 Z1 Z1 Z1 Z1#2 Z1#2 X.10 + 2 2 0 0 -2 -2 0 0 Z1 Z1#2 Z1#2 Z1 -Z1 -Z1-Z1#2-Z1#2 X.11 + 2 2 0 0 -2 -2 0 0 Z1#2 Z1 Z1 Z1#2-Z1#2-Z1#2 -Z1 -Z1 X.12 + 2 2 0 0 2 2 0 0 Z1#2 Z1 Z1 Z1#2 Z1#2 Z1#2 Z1 Z1 X.13 0 2 -2 0 0 2*I -2*I 0 0 Z1#2 Z1 -Z1-Z1#2 Z2 -Z2 Z2#3-Z2#3 X.14 0 2 -2 0 0 2*I -2*I 0 0 Z1 Z1#2-Z1#2 -Z1-Z2#3 Z2#3 -Z2 Z2 X.15 0 2 -2 0 0 -2*I 2*I 0 0 Z1#2 Z1 -Z1-Z1#2 -Z2 Z2-Z2#3 Z2#3 X.16 0 2 -2 0 0 -2*I 2*I 0 0 Z1 Z1#2-Z1#2 -Z1 Z2#3-Z2#3 Z2 -Z2 Explanation of Character Value Symbols -------------------------------------- # denotes algebraic conjugation, that is, #k indicates replacing the root of unity w by w^k I = RootOfUnity(4) Z1 = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | 0, 0, 1, 1 ] Z2 = (CyclotomicField(20: Sparse := true)) ! [ RationalField() | 0, -1, 0, 0, 0, -1, 0, -1 ] Conjugacy Classes of group G ---------------------------- [1] Order 1 Length 1 Rep Id(G) [2] Order 2 Length 1 Rep G.3 [3] Order 2 Length 5 Rep G.1 * G.3 [4] Order 2 Length 5 Rep G.1 [5] Order 4 Length 1 Rep G.2 [6] Order 4 Length 1 Rep G.2 * G.3 [7] Order 4 Length 5 Rep G.1 * G.2 * G.3 [8] Order 4 Length 5 Rep G.1 * G.2 [9] Order 5 Length 2 Rep G.4^2 [10] Order 5 Length 2 Rep G.4 [11] Order 10 Length 2 Rep G.3 * G.4 [12] Order 10 Length 2 Rep G.3 * G.4^2 [13] Order 20 Length 2 Rep G.2 * G.4^2 [14] Order 20 Length 2 Rep G.2 * G.3 * G.4^2 [15] Order 20 Length 2 Rep G.2 * G.3 * G.4 [16] Order 20 Length 2 Rep G.2 * G.4 Surface kernel generators: [ G.1, G.1 * G.2 * G.4^2, G.2 * G.3 * G.4^3 ] Is hyperelliptic? true Curve is hyperelliptic > FP,f:=FPGroup(G); > PermG,g:=PermutationGroup(FP); > PermG; Permutation group PermG acting on a set of cardinality 20 Order = 40 = 2^3 * 5 (4, 8)(6, 11)(7, 13)(9, 15)(10, 16)(12, 18)(14, 19)(17, 20) (1, 2, 3, 5)(4, 6, 7, 10)(8, 11, 13, 16)(9, 12, 14, 17)(15, 18, 19, 20) (1, 3)(2, 5)(4, 7)(6, 10)(8, 13)(9, 14)(11, 16)(12, 17)(15, 19)(18, 20) (1, 4, 9, 15, 8)(2, 6, 12, 18, 11)(3, 7, 14, 19, 13)(5, 10, 17, 20, 16) > g(Inverse(f)(G.1)); (4, 8)(6, 11)(7, 13)(9, 15)(10, 16)(12, 18)(14, 19)(17, 20) > g(Inverse(f)(G.1 * G.2 * G.4^2)); (1, 12, 3, 17)(2, 14, 5, 9)(4, 6, 7, 10)(8, 18, 13, 20)(11, 19, 16, 15)