Magma V2.21-7 Tue Mar 22 2016 07:07:12 on Davids-MacBook-Pro-2 [Seed = 680483090] +-------------------------------------------------------------------+ | This copy of Magma has been made available through a | | generous initiative of the | | | | Simons Foundation | | | | covering U.S. Colleges, Universities, Nonprofit Research entities,| | and their students, faculty, and staff | +-------------------------------------------------------------------+ Type ? for help. Type -D to quit. > load "autcv10d.txt"; Loading "autcv10d.txt" > G:=SmallGroup(56,7); > RunExample(G,6,[2,4,14]); 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 2 14 14 2 2 2 2 2 2 2 2 2 2 2 Order | 1 2 2 2 4 7 7 7 14 14 14 14 14 14 14 14 ------------------------------------------------------------------------------ p = 2 1 1 1 1 2 7 8 6 8 8 6 7 7 6 8 6 p = 7 1 2 3 4 5 1 1 1 2 3 3 3 3 3 3 2 ------------------------------------------------------------------------------ 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 + 2 -2 0 0 0 2 2 2 -2 0 0 0 0 0 0 -2 X.6 + 2 2 2 0 0 Z1 Z1#2 Z1#3 Z1#2 Z1#2 Z1#3 Z1 Z1 Z1#3 Z1#2 Z1#3 X.7 + 2 2 2 0 0 Z1#2 Z1#3 Z1 Z1#3 Z1#3 Z1 Z1#2 Z1#2 Z1 Z1#3 Z1 X.8 0 2 -2 0 0 0 Z1#2 Z1#3 Z1-Z1#3 Z2 Z2#2 Z2#3-Z2#3-Z2#2 -Z2 -Z1 X.9 0 2 -2 0 0 0 Z1 Z1#2 Z1#3-Z1#2 Z2#3 -Z2 Z2#2-Z2#2 Z2-Z2#3-Z1#3 X.10 + 2 2 -2 0 0 Z1#3 Z1 Z1#2 Z1 -Z1-Z1#2-Z1#3-Z1#3-Z1#2 -Z1 Z1#2 X.11 + 2 2 -2 0 0 Z1#2 Z1#3 Z1 Z1#3-Z1#3 -Z1-Z1#2-Z1#2 -Z1-Z1#3 Z1 X.12 + 2 2 2 0 0 Z1#3 Z1 Z1#2 Z1 Z1 Z1#2 Z1#3 Z1#3 Z1#2 Z1 Z1#2 X.13 0 2 -2 0 0 0 Z1#3 Z1 Z1#2 -Z1-Z2#2 Z2#3 Z2 -Z2-Z2#3 Z2#2-Z1#2 X.14 + 2 2 -2 0 0 Z1 Z1#2 Z1#3 Z1#2-Z1#2-Z1#3 -Z1 -Z1-Z1#3-Z1#2 Z1#3 X.15 0 2 -2 0 0 0 Z1 Z1#2 Z1#3-Z1#2-Z2#3 Z2-Z2#2 Z2#2 -Z2 Z2#3-Z1#3 X.16 0 2 -2 0 0 0 Z1#2 Z1#3 Z1-Z1#3 -Z2-Z2#2-Z2#3 Z2#3 Z2#2 Z2 -Z1 X.17 0 2 -2 0 0 0 Z1#3 Z1 Z1#2 -Z1 Z2#2-Z2#3 -Z2 Z2 Z2#3-Z2#2-Z1#2 -------------- Class | 17 Size | 2 Order | 14 -------------- p = 2 7 p = 7 2 -------------- X.1 + 1 X.2 + 1 X.3 + 1 X.4 + 1 X.5 + -2 X.6 + Z1 X.7 + Z1#2 X.8 0 -Z1#2 X.9 0 -Z1 X.10 + Z1#3 X.11 + Z1#2 X.12 + Z1#3 X.13 0 -Z1#3 X.14 + Z1 X.15 0 -Z1 X.16 0 -Z1#2 X.17 0 -Z1#3 Explanation of Character Value Symbols -------------------------------------- # denotes algebraic conjugation, that is, #k indicates replacing the root of unity w by w^k Z1 = (CyclotomicField(7: Sparse := true)) ! [ RationalField() | 0, 0, 0, 1, 1, 0 ] Z2 = (CyclotomicField(7: Sparse := true)) ! [ RationalField() | 0, 0, -1, 0, 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 2 Rep G.2 [4] Order 2 Length 14 Rep G.1 [5] Order 4 Length 14 Rep G.1 * G.2 * G.3 [6] Order 7 Length 2 Rep G.4^3 [7] Order 7 Length 2 Rep G.4 [8] Order 7 Length 2 Rep G.4^2 [9] Order 14 Length 2 Rep G.3 * G.4 [10] Order 14 Length 2 Rep G.2 * G.4 [11] Order 14 Length 2 Rep G.2 * G.4^2 [12] Order 14 Length 2 Rep G.2 * G.4^3 [13] Order 14 Length 2 Rep G.2 * G.4^4 [14] Order 14 Length 2 Rep G.2 * G.4^5 [15] Order 14 Length 2 Rep G.2 * G.4^6 [16] Order 14 Length 2 Rep G.3 * G.4^2 [17] Order 14 Length 2 Rep G.3 * G.4^3 Surface kernel generators: [ G.1 * G.4^3, G.1 * G.2 * G.4^2, G.2 * G.4 ] Is hyperelliptic? true Curve is hyperelliptic > FP,f:=FPGroup(G); > PermG,g:=PermutationGroup(FP); > g(Inverse(f)(G.1 * G.4^3)); (1, 17)(2, 25)(3, 22)(4, 9)(5, 20)(6, 18)(7, 14)(8, 24)(10, 12)(11, 27)(13, 28)(15, 26)(19, 23) > g(Inverse(f)(G.1 * G.2 * G.4^2)); (1, 12, 3, 18)(2, 14, 5, 9)(4, 6, 7, 10)(8, 20, 13, 25)(11, 17, 15, 22)(16, 27, 21, 26)(19, 24, 23, 28)