direct product, cyclic, abelian, monomial
Aliases: C126, also denoted Z126, SmallGroup(126,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C126 |
| C1 — C126 |
| C1 — C126 |
Generators and relations for C126
G = < a | a126=1 >
Smallest permutation representation of C126
►Regular action on 126 points
Generators in S126
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)
G:=sub<Sym(126)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126)]])
C126 is a maximal subgroup of
Dic63
126 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | ··· | 7F | 9A | ··· | 9F | 14A | ··· | 14F | 18A | ··· | 18F | 21A | ··· | 21L | 42A | ··· | 42L | 63A | ··· | 63AJ | 126A | ··· | 126AJ |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 9 | ··· | 9 | 14 | ··· | 14 | 18 | ··· | 18 | 21 | ··· | 21 | 42 | ··· | 42 | 63 | ··· | 63 | 126 | ··· | 126 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | C7 | C9 | C14 | C18 | C21 | C42 | C63 | C126 |
| kernel | C126 | C63 | C42 | C21 | C18 | C14 | C9 | C7 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 36 | 36 |
Matrix representation of C126 ►in GL1(𝔽127) generated by
| 106 |
G:=sub<GL(1,GF(127))| [106] >;
C126 in GAP, Magma, Sage, TeX
C_{126} % in TeX
G:=Group("C126"); // GroupNames label
G:=SmallGroup(126,6);
// by ID
G=gap.SmallGroup(126,6);
# by ID
G:=PCGroup([4,-2,-3,-7,-3,173]);
// Polycyclic
G:=Group<a|a^126=1>;
// generators/relations
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