direct product, abelian, monomial, 2-elementary
Aliases: C2×C132, SmallGroup(264,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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
(1 263)(2 264)(3 133)(4 134)(5 135)(6 136)(7 137)(8 138)(9 139)(10 140)(11 141)(12 142)(13 143)(14 144)(15 145)(16 146)(17 147)(18 148)(19 149)(20 150)(21 151)(22 152)(23 153)(24 154)(25 155)(26 156)(27 157)(28 158)(29 159)(30 160)(31 161)(32 162)(33 163)(34 164)(35 165)(36 166)(37 167)(38 168)(39 169)(40 170)(41 171)(42 172)(43 173)(44 174)(45 175)(46 176)(47 177)(48 178)(49 179)(50 180)(51 181)(52 182)(53 183)(54 184)(55 185)(56 186)(57 187)(58 188)(59 189)(60 190)(61 191)(62 192)(63 193)(64 194)(65 195)(66 196)(67 197)(68 198)(69 199)(70 200)(71 201)(72 202)(73 203)(74 204)(75 205)(76 206)(77 207)(78 208)(79 209)(80 210)(81 211)(82 212)(83 213)(84 214)(85 215)(86 216)(87 217)(88 218)(89 219)(90 220)(91 221)(92 222)(93 223)(94 224)(95 225)(96 226)(97 227)(98 228)(99 229)(100 230)(101 231)(102 232)(103 233)(104 234)(105 235)(106 236)(107 237)(108 238)(109 239)(110 240)(111 241)(112 242)(113 243)(114 244)(115 245)(116 246)(117 247)(118 248)(119 249)(120 250)(121 251)(122 252)(123 253)(124 254)(125 255)(126 256)(127 257)(128 258)(129 259)(130 260)(131 261)(132 262)
(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)| (1,263)(2,264)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,141)(12,142)(13,143)(14,144)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,175)(46,176)(47,177)(48,178)(49,179)(50,180)(51,181)(52,182)(53,183)(54,184)(55,185)(56,186)(57,187)(58,188)(59,189)(60,190)(61,191)(62,192)(63,193)(64,194)(65,195)(66,196)(67,197)(68,198)(69,199)(70,200)(71,201)(72,202)(73,203)(74,204)(75,205)(76,206)(77,207)(78,208)(79,209)(80,210)(81,211)(82,212)(83,213)(84,214)(85,215)(86,216)(87,217)(88,218)(89,219)(90,220)(91,221)(92,222)(93,223)(94,224)(95,225)(96,226)(97,227)(98,228)(99,229)(100,230)(101,231)(102,232)(103,233)(104,234)(105,235)(106,236)(107,237)(108,238)(109,239)(110,240)(111,241)(112,242)(113,243)(114,244)(115,245)(116,246)(117,247)(118,248)(119,249)(120,250)(121,251)(122,252)(123,253)(124,254)(125,255)(126,256)(127,257)(128,258)(129,259)(130,260)(131,261)(132,262), (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( (1,263)(2,264)(3,133)(4,134)(5,135)(6,136)(7,137)(8,138)(9,139)(10,140)(11,141)(12,142)(13,143)(14,144)(15,145)(16,146)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,154)(25,155)(26,156)(27,157)(28,158)(29,159)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,175)(46,176)(47,177)(48,178)(49,179)(50,180)(51,181)(52,182)(53,183)(54,184)(55,185)(56,186)(57,187)(58,188)(59,189)(60,190)(61,191)(62,192)(63,193)(64,194)(65,195)(66,196)(67,197)(68,198)(69,199)(70,200)(71,201)(72,202)(73,203)(74,204)(75,205)(76,206)(77,207)(78,208)(79,209)(80,210)(81,211)(82,212)(83,213)(84,214)(85,215)(86,216)(87,217)(88,218)(89,219)(90,220)(91,221)(92,222)(93,223)(94,224)(95,225)(96,226)(97,227)(98,228)(99,229)(100,230)(101,231)(102,232)(103,233)(104,234)(105,235)(106,236)(107,237)(108,238)(109,239)(110,240)(111,241)(112,242)(113,243)(114,244)(115,245)(116,246)(117,247)(118,248)(119,249)(120,250)(121,251)(122,252)(123,253)(124,254)(125,255)(126,256)(127,257)(128,258)(129,259)(130,260)(131,261)(132,262), (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([[(1,263),(2,264),(3,133),(4,134),(5,135),(6,136),(7,137),(8,138),(9,139),(10,140),(11,141),(12,142),(13,143),(14,144),(15,145),(16,146),(17,147),(18,148),(19,149),(20,150),(21,151),(22,152),(23,153),(24,154),(25,155),(26,156),(27,157),(28,158),(29,159),(30,160),(31,161),(32,162),(33,163),(34,164),(35,165),(36,166),(37,167),(38,168),(39,169),(40,170),(41,171),(42,172),(43,173),(44,174),(45,175),(46,176),(47,177),(48,178),(49,179),(50,180),(51,181),(52,182),(53,183),(54,184),(55,185),(56,186),(57,187),(58,188),(59,189),(60,190),(61,191),(62,192),(63,193),(64,194),(65,195),(66,196),(67,197),(68,198),(69,199),(70,200),(71,201),(72,202),(73,203),(74,204),(75,205),(76,206),(77,207),(78,208),(79,209),(80,210),(81,211),(82,212),(83,213),(84,214),(85,215),(86,216),(87,217),(88,218),(89,219),(90,220),(91,221),(92,222),(93,223),(94,224),(95,225),(96,226),(97,227),(98,228),(99,229),(100,230),(101,231),(102,232),(103,233),(104,234),(105,235),(106,236),(107,237),(108,238),(109,239),(110,240),(111,241),(112,242),(113,243),(114,244),(115,245),(116,246),(117,247),(118,248),(119,249),(120,250),(121,251),(122,252),(123,253),(124,254),(125,255),(126,256),(127,257),(128,258),(129,259),(130,260),(131,261),(132,262)], [(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 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 11A | ··· | 11J | 12A | ··· | 12H | 22A | ··· | 22AD | 33A | ··· | 33T | 44A | ··· | 44AN | 66A | ··· | 66BH | 132A | ··· | 132CB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 11 | ··· | 11 | 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 | 1 | ··· | 1 |
264 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C11 | C12 | C22 | C22 | C33 | C44 | C66 | C66 | C132 |
| kernel | C2×C132 | C132 | C2×C66 | C2×C44 | C66 | C44 | C2×C22 | C2×C12 | C22 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 10 | 8 | 20 | 10 | 20 | 40 | 40 | 20 | 80 |
Matrix representation of C2×C132 ►in GL2(𝔽397) generated by
| 396 | 0 |
| 0 | 396 |
| 64 | 0 |
| 0 | 267 |
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
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