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G = C132order 169 = 132

Elementary abelian group of type [13,13]

direct product, p-group, elementary abelian, monomial

Aliases: C132, SmallGroup(169,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C132
C1C13 — C132
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 98 84 69 133 127 36 60 107 46 162 18 150)(2 99 85 70 134 128 37 61 108 47 163 19 151)(3 100 86 71 135 129 38 62 109 48 164 20 152)(4 101 87 72 136 130 39 63 110 49 165 21 153)(5 102 88 73 137 118 27 64 111 50 166 22 154)(6 103 89 74 138 119 28 65 112 51 167 23 155)(7 104 90 75 139 120 29 53 113 52 168 24 156)(8 92 91 76 140 121 30 54 114 40 169 25 144)(9 93 79 77 141 122 31 55 115 41 157 26 145)(10 94 80 78 142 123 32 56 116 42 158 14 146)(11 95 81 66 143 124 33 57 117 43 159 15 147)(12 96 82 67 131 125 34 58 105 44 160 16 148)(13 97 83 68 132 126 35 59 106 45 161 17 149)

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), (1,98,84,69,133,127,36,60,107,46,162,18,150)(2,99,85,70,134,128,37,61,108,47,163,19,151)(3,100,86,71,135,129,38,62,109,48,164,20,152)(4,101,87,72,136,130,39,63,110,49,165,21,153)(5,102,88,73,137,118,27,64,111,50,166,22,154)(6,103,89,74,138,119,28,65,112,51,167,23,155)(7,104,90,75,139,120,29,53,113,52,168,24,156)(8,92,91,76,140,121,30,54,114,40,169,25,144)(9,93,79,77,141,122,31,55,115,41,157,26,145)(10,94,80,78,142,123,32,56,116,42,158,14,146)(11,95,81,66,143,124,33,57,117,43,159,15,147)(12,96,82,67,131,125,34,58,105,44,160,16,148)(13,97,83,68,132,126,35,59,106,45,161,17,149)>;

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), (1,98,84,69,133,127,36,60,107,46,162,18,150)(2,99,85,70,134,128,37,61,108,47,163,19,151)(3,100,86,71,135,129,38,62,109,48,164,20,152)(4,101,87,72,136,130,39,63,110,49,165,21,153)(5,102,88,73,137,118,27,64,111,50,166,22,154)(6,103,89,74,138,119,28,65,112,51,167,23,155)(7,104,90,75,139,120,29,53,113,52,168,24,156)(8,92,91,76,140,121,30,54,114,40,169,25,144)(9,93,79,77,141,122,31,55,115,41,157,26,145)(10,94,80,78,142,123,32,56,116,42,158,14,146)(11,95,81,66,143,124,33,57,117,43,159,15,147)(12,96,82,67,131,125,34,58,105,44,160,16,148)(13,97,83,68,132,126,35,59,106,45,161,17,149) );

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)], [(1,98,84,69,133,127,36,60,107,46,162,18,150),(2,99,85,70,134,128,37,61,108,47,163,19,151),(3,100,86,71,135,129,38,62,109,48,164,20,152),(4,101,87,72,136,130,39,63,110,49,165,21,153),(5,102,88,73,137,118,27,64,111,50,166,22,154),(6,103,89,74,138,119,28,65,112,51,167,23,155),(7,104,90,75,139,120,29,53,113,52,168,24,156),(8,92,91,76,140,121,30,54,114,40,169,25,144),(9,93,79,77,141,122,31,55,115,41,157,26,145),(10,94,80,78,142,123,32,56,116,42,158,14,146),(11,95,81,66,143,124,33,57,117,43,159,15,147),(12,96,82,67,131,125,34,58,105,44,160,16,148),(13,97,83,68,132,126,35,59,106,45,161,17,149)])

C132 is a maximal subgroup of   C13⋊D13

169 conjugacy classes

class 1 13A···13FL
order113···13
size11···1

169 irreducible representations

dim11
type+
imageC1C13
kernelC132C13
# reps1168

Matrix representation of C132 in GL2(𝔽53) generated by

130
01
,
280
028
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

Export

Subgroup lattice of C132 in TeX

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