Copied to
clipboard

G = C2×C132order 264 = 23·3·11

Abelian group of type [2,132]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C132, SmallGroup(264,28)

Series: Derived Chief Lower central Upper central

C1 — C2×C132
C1C2C22C66C132 — C2×C132
C1 — C2×C132
C1 — C2×C132

Generators and relations for C2×C132
 G = < a,b | a2=b132=1, ab=ba >


Smallest permutation representation of C2×C132
Regular action on 264 points
Generators in S264
(1 147)(2 148)(3 149)(4 150)(5 151)(6 152)(7 153)(8 154)(9 155)(10 156)(11 157)(12 158)(13 159)(14 160)(15 161)(16 162)(17 163)(18 164)(19 165)(20 166)(21 167)(22 168)(23 169)(24 170)(25 171)(26 172)(27 173)(28 174)(29 175)(30 176)(31 177)(32 178)(33 179)(34 180)(35 181)(36 182)(37 183)(38 184)(39 185)(40 186)(41 187)(42 188)(43 189)(44 190)(45 191)(46 192)(47 193)(48 194)(49 195)(50 196)(51 197)(52 198)(53 199)(54 200)(55 201)(56 202)(57 203)(58 204)(59 205)(60 206)(61 207)(62 208)(63 209)(64 210)(65 211)(66 212)(67 213)(68 214)(69 215)(70 216)(71 217)(72 218)(73 219)(74 220)(75 221)(76 222)(77 223)(78 224)(79 225)(80 226)(81 227)(82 228)(83 229)(84 230)(85 231)(86 232)(87 233)(88 234)(89 235)(90 236)(91 237)(92 238)(93 239)(94 240)(95 241)(96 242)(97 243)(98 244)(99 245)(100 246)(101 247)(102 248)(103 249)(104 250)(105 251)(106 252)(107 253)(108 254)(109 255)(110 256)(111 257)(112 258)(113 259)(114 260)(115 261)(116 262)(117 263)(118 264)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(127 141)(128 142)(129 143)(130 144)(131 145)(132 146)
(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)(133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264)

G:=sub<Sym(264)| (1,147)(2,148)(3,149)(4,150)(5,151)(6,152)(7,153)(8,154)(9,155)(10,156)(11,157)(12,158)(13,159)(14,160)(15,161)(16,162)(17,163)(18,164)(19,165)(20,166)(21,167)(22,168)(23,169)(24,170)(25,171)(26,172)(27,173)(28,174)(29,175)(30,176)(31,177)(32,178)(33,179)(34,180)(35,181)(36,182)(37,183)(38,184)(39,185)(40,186)(41,187)(42,188)(43,189)(44,190)(45,191)(46,192)(47,193)(48,194)(49,195)(50,196)(51,197)(52,198)(53,199)(54,200)(55,201)(56,202)(57,203)(58,204)(59,205)(60,206)(61,207)(62,208)(63,209)(64,210)(65,211)(66,212)(67,213)(68,214)(69,215)(70,216)(71,217)(72,218)(73,219)(74,220)(75,221)(76,222)(77,223)(78,224)(79,225)(80,226)(81,227)(82,228)(83,229)(84,230)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,241)(96,242)(97,243)(98,244)(99,245)(100,246)(101,247)(102,248)(103,249)(104,250)(105,251)(106,252)(107,253)(108,254)(109,255)(110,256)(111,257)(112,258)(113,259)(114,260)(115,261)(116,262)(117,263)(118,264)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(129,143)(130,144)(131,145)(132,146), (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)(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264)>;

G:=Group( (1,147)(2,148)(3,149)(4,150)(5,151)(6,152)(7,153)(8,154)(9,155)(10,156)(11,157)(12,158)(13,159)(14,160)(15,161)(16,162)(17,163)(18,164)(19,165)(20,166)(21,167)(22,168)(23,169)(24,170)(25,171)(26,172)(27,173)(28,174)(29,175)(30,176)(31,177)(32,178)(33,179)(34,180)(35,181)(36,182)(37,183)(38,184)(39,185)(40,186)(41,187)(42,188)(43,189)(44,190)(45,191)(46,192)(47,193)(48,194)(49,195)(50,196)(51,197)(52,198)(53,199)(54,200)(55,201)(56,202)(57,203)(58,204)(59,205)(60,206)(61,207)(62,208)(63,209)(64,210)(65,211)(66,212)(67,213)(68,214)(69,215)(70,216)(71,217)(72,218)(73,219)(74,220)(75,221)(76,222)(77,223)(78,224)(79,225)(80,226)(81,227)(82,228)(83,229)(84,230)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,241)(96,242)(97,243)(98,244)(99,245)(100,246)(101,247)(102,248)(103,249)(104,250)(105,251)(106,252)(107,253)(108,254)(109,255)(110,256)(111,257)(112,258)(113,259)(114,260)(115,261)(116,262)(117,263)(118,264)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(129,143)(130,144)(131,145)(132,146), (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)(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264) );

G=PermutationGroup([(1,147),(2,148),(3,149),(4,150),(5,151),(6,152),(7,153),(8,154),(9,155),(10,156),(11,157),(12,158),(13,159),(14,160),(15,161),(16,162),(17,163),(18,164),(19,165),(20,166),(21,167),(22,168),(23,169),(24,170),(25,171),(26,172),(27,173),(28,174),(29,175),(30,176),(31,177),(32,178),(33,179),(34,180),(35,181),(36,182),(37,183),(38,184),(39,185),(40,186),(41,187),(42,188),(43,189),(44,190),(45,191),(46,192),(47,193),(48,194),(49,195),(50,196),(51,197),(52,198),(53,199),(54,200),(55,201),(56,202),(57,203),(58,204),(59,205),(60,206),(61,207),(62,208),(63,209),(64,210),(65,211),(66,212),(67,213),(68,214),(69,215),(70,216),(71,217),(72,218),(73,219),(74,220),(75,221),(76,222),(77,223),(78,224),(79,225),(80,226),(81,227),(82,228),(83,229),(84,230),(85,231),(86,232),(87,233),(88,234),(89,235),(90,236),(91,237),(92,238),(93,239),(94,240),(95,241),(96,242),(97,243),(98,244),(99,245),(100,246),(101,247),(102,248),(103,249),(104,250),(105,251),(106,252),(107,253),(108,254),(109,255),(110,256),(111,257),(112,258),(113,259),(114,260),(115,261),(116,262),(117,263),(118,264),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(127,141),(128,142),(129,143),(130,144),(131,145),(132,146)], [(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),(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264)])

264 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F11A···11J12A···12H22A···22AD33A···33T44A···44AN66A···66BH132A···132CB
order12223344446···611···1112···1222···2233···3344···4466···66132···132
size11111111111···11···11···11···11···11···11···11···1

264 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C4C6C6C11C12C22C22C33C44C66C66C132
kernelC2×C132C132C2×C66C2×C44C66C44C2×C22C2×C12C22C12C2×C6C2×C4C6C4C22C2
# reps121244210820102040402080

Matrix representation of C2×C132 in GL2(𝔽397) generated by

3960
0396
,
640
0267
G:=sub<GL(2,GF(397))| [396,0,0,396],[64,0,0,267] >;

C2×C132 in GAP, Magma, Sage, TeX

C_2\times C_{132}
% in TeX

G:=Group("C2xC132");
// GroupNames label

G:=SmallGroup(264,28);
// by ID

G=gap.SmallGroup(264,28);
# by ID

G:=PCGroup([5,-2,-2,-3,-11,-2,660]);
// Polycyclic

G:=Group<a,b|a^2=b^132=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C132 in TeX

׿
×
𝔽