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G = C2×C116order 232 = 23·29

Abelian group of type [2,116]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C116, SmallGroup(232,9)

Series: Derived Chief Lower central Upper central

C1 — C2×C116
C1C2C58C116 — C2×C116
C1 — C2×C116
C1 — C2×C116

Generators and relations for C2×C116
 G = < a,b | a2=b116=1, ab=ba >


Smallest permutation representation of C2×C116
Regular action on 232 points
Generators in S232
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 161)(19 162)(20 163)(21 164)(22 165)(23 166)(24 167)(25 168)(26 169)(27 170)(28 171)(29 172)(30 173)(31 174)(32 175)(33 176)(34 177)(35 178)(36 179)(37 180)(38 181)(39 182)(40 183)(41 184)(42 185)(43 186)(44 187)(45 188)(46 189)(47 190)(48 191)(49 192)(50 193)(51 194)(52 195)(53 196)(54 197)(55 198)(56 199)(57 200)(58 201)(59 202)(60 203)(61 204)(62 205)(63 206)(64 207)(65 208)(66 209)(67 210)(68 211)(69 212)(70 213)(71 214)(72 215)(73 216)(74 217)(75 218)(76 219)(77 220)(78 221)(79 222)(80 223)(81 224)(82 225)(83 226)(84 227)(85 228)(86 229)(87 230)(88 231)(89 232)(90 117)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(111 138)(112 139)(113 140)(114 141)(115 142)(116 143)
(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)(117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232)

G:=sub<Sym(232)| (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,160)(18,161)(19,162)(20,163)(21,164)(22,165)(23,166)(24,167)(25,168)(26,169)(27,170)(28,171)(29,172)(30,173)(31,174)(32,175)(33,176)(34,177)(35,178)(36,179)(37,180)(38,181)(39,182)(40,183)(41,184)(42,185)(43,186)(44,187)(45,188)(46,189)(47,190)(48,191)(49,192)(50,193)(51,194)(52,195)(53,196)(54,197)(55,198)(56,199)(57,200)(58,201)(59,202)(60,203)(61,204)(62,205)(63,206)(64,207)(65,208)(66,209)(67,210)(68,211)(69,212)(70,213)(71,214)(72,215)(73,216)(74,217)(75,218)(76,219)(77,220)(78,221)(79,222)(80,223)(81,224)(82,225)(83,226)(84,227)(85,228)(86,229)(87,230)(88,231)(89,232)(90,117)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143), (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)(117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232)>;

G:=Group( (1,144)(2,145)(3,146)(4,147)(5,148)(6,149)(7,150)(8,151)(9,152)(10,153)(11,154)(12,155)(13,156)(14,157)(15,158)(16,159)(17,160)(18,161)(19,162)(20,163)(21,164)(22,165)(23,166)(24,167)(25,168)(26,169)(27,170)(28,171)(29,172)(30,173)(31,174)(32,175)(33,176)(34,177)(35,178)(36,179)(37,180)(38,181)(39,182)(40,183)(41,184)(42,185)(43,186)(44,187)(45,188)(46,189)(47,190)(48,191)(49,192)(50,193)(51,194)(52,195)(53,196)(54,197)(55,198)(56,199)(57,200)(58,201)(59,202)(60,203)(61,204)(62,205)(63,206)(64,207)(65,208)(66,209)(67,210)(68,211)(69,212)(70,213)(71,214)(72,215)(73,216)(74,217)(75,218)(76,219)(77,220)(78,221)(79,222)(80,223)(81,224)(82,225)(83,226)(84,227)(85,228)(86,229)(87,230)(88,231)(89,232)(90,117)(91,118)(92,119)(93,120)(94,121)(95,122)(96,123)(97,124)(98,125)(99,126)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143), (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)(117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232) );

G=PermutationGroup([(1,144),(2,145),(3,146),(4,147),(5,148),(6,149),(7,150),(8,151),(9,152),(10,153),(11,154),(12,155),(13,156),(14,157),(15,158),(16,159),(17,160),(18,161),(19,162),(20,163),(21,164),(22,165),(23,166),(24,167),(25,168),(26,169),(27,170),(28,171),(29,172),(30,173),(31,174),(32,175),(33,176),(34,177),(35,178),(36,179),(37,180),(38,181),(39,182),(40,183),(41,184),(42,185),(43,186),(44,187),(45,188),(46,189),(47,190),(48,191),(49,192),(50,193),(51,194),(52,195),(53,196),(54,197),(55,198),(56,199),(57,200),(58,201),(59,202),(60,203),(61,204),(62,205),(63,206),(64,207),(65,208),(66,209),(67,210),(68,211),(69,212),(70,213),(71,214),(72,215),(73,216),(74,217),(75,218),(76,219),(77,220),(78,221),(79,222),(80,223),(81,224),(82,225),(83,226),(84,227),(85,228),(86,229),(87,230),(88,231),(89,232),(90,117),(91,118),(92,119),(93,120),(94,121),(95,122),(96,123),(97,124),(98,125),(99,126),(100,127),(101,128),(102,129),(103,130),(104,131),(105,132),(106,133),(107,134),(108,135),(109,136),(110,137),(111,138),(112,139),(113,140),(114,141),(115,142),(116,143)], [(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),(117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232)])

C2×C116 is a maximal subgroup of   C4.Dic29  C58.D4  C4⋊Dic29  D58⋊C4  D1165C2

232 conjugacy classes

class 1 2A2B2C4A4B4C4D29A···29AB58A···58CF116A···116DH
order1222444429···2958···58116···116
size111111111···11···11···1

232 irreducible representations

dim11111111
type+++
imageC1C2C2C4C29C58C58C116
kernelC2×C116C116C2×C58C58C2×C4C4C22C2
# reps1214285628112

Matrix representation of C2×C116 in GL2(𝔽233) generated by

2320
01
,
130
0202
G:=sub<GL(2,GF(233))| [232,0,0,1],[13,0,0,202] >;

C2×C116 in GAP, Magma, Sage, TeX

C_2\times C_{116}
% in TeX

G:=Group("C2xC116");
// GroupNames label

G:=SmallGroup(232,9);
// by ID

G=gap.SmallGroup(232,9);
# by ID

G:=PCGroup([4,-2,-2,-29,-2,464]);
// Polycyclic

G:=Group<a,b|a^2=b^116=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C116 in TeX

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