direct product, cyclic, abelian, monomial
Aliases: C132, also denoted Z132, SmallGroup(132,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C132 |
| C1 — C132 |
| C1 — C132 |
Generators and relations for C132
G = < a | a132=1 >
Smallest permutation representation of C132
►Regular action on 132 points
Generators in S132
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132)
G:=sub<Sym(132)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132)]])
C132 is a maximal subgroup of
C33⋊C8 Dic66 D132
132 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 11A | ··· | 11J | 12A | 12B | 12C | 12D | 22A | ··· | 22J | 33A | ··· | 33T | 44A | ··· | 44T | 66A | ··· | 66T | 132A | ··· | 132AN |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 11 | ··· | 11 | 12 | 12 | 12 | 12 | 22 | ··· | 22 | 33 | ··· | 33 | 44 | ··· | 44 | 66 | ··· | 66 | 132 | ··· | 132 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C11 | C12 | C22 | C33 | C44 | C66 | C132 |
| kernel | C132 | C66 | C44 | C33 | C22 | C12 | C11 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 10 | 4 | 10 | 20 | 20 | 20 | 40 |
Matrix representation of C132 ►in GL2(𝔽23) generated by
| 0 | 7 |
| 1 | 18 |
G:=sub<GL(2,GF(23))| [0,1,7,18] >;
C132 in GAP, Magma, Sage, TeX
C_{132} % in TeX
G:=Group("C132"); // GroupNames label
G:=SmallGroup(132,4);
// by ID
G=gap.SmallGroup(132,4);
# by ID
G:=PCGroup([4,-2,-3,-11,-2,264]);
// Polycyclic
G:=Group<a|a^132=1>;
// generators/relations
Export