direct product, p-group, elementary abelian, monomial
Aliases: C132, SmallGroup(169,2)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C132 |
| C1 — C132 |
| C1 — C132 |
Generators and relations for C132
G = < a,b | a13=b13=1, ab=ba >
Smallest permutation representation of C132
►Regular action on 169 points
Generators in S169
(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)(131 132 133 134 135 136 137 138 139 140 141 142 143)(144 145 146 147 148 149 150 151 152 153 154 155 156)(157 158 159 160 161 162 163 164 165 166 167 168 169)
(1 97 41 113 65 119 19 32 136 81 167 144 71)(2 98 42 114 53 120 20 33 137 82 168 145 72)(3 99 43 115 54 121 21 34 138 83 169 146 73)(4 100 44 116 55 122 22 35 139 84 157 147 74)(5 101 45 117 56 123 23 36 140 85 158 148 75)(6 102 46 105 57 124 24 37 141 86 159 149 76)(7 103 47 106 58 125 25 38 142 87 160 150 77)(8 104 48 107 59 126 26 39 143 88 161 151 78)(9 92 49 108 60 127 14 27 131 89 162 152 66)(10 93 50 109 61 128 15 28 132 90 163 153 67)(11 94 51 110 62 129 16 29 133 91 164 154 68)(12 95 52 111 63 130 17 30 134 79 165 155 69)(13 96 40 112 64 118 18 31 135 80 166 156 70)
G:=sub<Sym(169)| (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)(131,132,133,134,135,136,137,138,139,140,141,142,143)(144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169), (1,97,41,113,65,119,19,32,136,81,167,144,71)(2,98,42,114,53,120,20,33,137,82,168,145,72)(3,99,43,115,54,121,21,34,138,83,169,146,73)(4,100,44,116,55,122,22,35,139,84,157,147,74)(5,101,45,117,56,123,23,36,140,85,158,148,75)(6,102,46,105,57,124,24,37,141,86,159,149,76)(7,103,47,106,58,125,25,38,142,87,160,150,77)(8,104,48,107,59,126,26,39,143,88,161,151,78)(9,92,49,108,60,127,14,27,131,89,162,152,66)(10,93,50,109,61,128,15,28,132,90,163,153,67)(11,94,51,110,62,129,16,29,133,91,164,154,68)(12,95,52,111,63,130,17,30,134,79,165,155,69)(13,96,40,112,64,118,18,31,135,80,166,156,70)>;
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)(131,132,133,134,135,136,137,138,139,140,141,142,143)(144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169), (1,97,41,113,65,119,19,32,136,81,167,144,71)(2,98,42,114,53,120,20,33,137,82,168,145,72)(3,99,43,115,54,121,21,34,138,83,169,146,73)(4,100,44,116,55,122,22,35,139,84,157,147,74)(5,101,45,117,56,123,23,36,140,85,158,148,75)(6,102,46,105,57,124,24,37,141,86,159,149,76)(7,103,47,106,58,125,25,38,142,87,160,150,77)(8,104,48,107,59,126,26,39,143,88,161,151,78)(9,92,49,108,60,127,14,27,131,89,162,152,66)(10,93,50,109,61,128,15,28,132,90,163,153,67)(11,94,51,110,62,129,16,29,133,91,164,154,68)(12,95,52,111,63,130,17,30,134,79,165,155,69)(13,96,40,112,64,118,18,31,135,80,166,156,70) );
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),(131,132,133,134,135,136,137,138,139,140,141,142,143),(144,145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168,169)], [(1,97,41,113,65,119,19,32,136,81,167,144,71),(2,98,42,114,53,120,20,33,137,82,168,145,72),(3,99,43,115,54,121,21,34,138,83,169,146,73),(4,100,44,116,55,122,22,35,139,84,157,147,74),(5,101,45,117,56,123,23,36,140,85,158,148,75),(6,102,46,105,57,124,24,37,141,86,159,149,76),(7,103,47,106,58,125,25,38,142,87,160,150,77),(8,104,48,107,59,126,26,39,143,88,161,151,78),(9,92,49,108,60,127,14,27,131,89,162,152,66),(10,93,50,109,61,128,15,28,132,90,163,153,67),(11,94,51,110,62,129,16,29,133,91,164,154,68),(12,95,52,111,63,130,17,30,134,79,165,155,69),(13,96,40,112,64,118,18,31,135,80,166,156,70)]])
C132 is a maximal subgroup of
C13⋊D13
169 conjugacy classes
| class | 1 | 13A | ··· | 13FL |
| order | 1 | 13 | ··· | 13 |
| size | 1 | 1 | ··· | 1 |
169 irreducible representations
| dim | 1 | 1 |
| type | + | |
| image | C1 | C13 |
| kernel | C132 | C13 |
| # reps | 1 | 168 |
Matrix representation of C132 ►in GL2(𝔽53) generated by
| 13 | 0 |
| 0 | 1 |
| 28 | 0 |
| 0 | 28 |
G:=sub<GL(2,GF(53))| [13,0,0,1],[28,0,0,28] >;
C132 in GAP, Magma, Sage, TeX
C_{13}^2 % in TeX
G:=Group("C13^2"); // GroupNames label
G:=SmallGroup(169,2);
// by ID
G=gap.SmallGroup(169,2);
# by ID
G:=PCGroup([2,-13,13]:ExponentLimit:=1);
// Polycyclic
G:=Group<a,b|a^13=b^13=1,a*b=b*a>;
// generators/relations
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