direct product, abelian, monomial, 2-elementary
Aliases: C2×C138, SmallGroup(276,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C138 |
| C1 — C2×C138 |
| C1 — C2×C138 |
Generators and relations for C2×C138
G = < a,b | a2=b138=1, ab=ba >
Smallest permutation representation of C2×C138
►Regular action on 276 points
Generators in S276
(1 242)(2 243)(3 244)(4 245)(5 246)(6 247)(7 248)(8 249)(9 250)(10 251)(11 252)(12 253)(13 254)(14 255)(15 256)(16 257)(17 258)(18 259)(19 260)(20 261)(21 262)(22 263)(23 264)(24 265)(25 266)(26 267)(27 268)(28 269)(29 270)(30 271)(31 272)(32 273)(33 274)(34 275)(35 276)(36 139)(37 140)(38 141)(39 142)(40 143)(41 144)(42 145)(43 146)(44 147)(45 148)(46 149)(47 150)(48 151)(49 152)(50 153)(51 154)(52 155)(53 156)(54 157)(55 158)(56 159)(57 160)(58 161)(59 162)(60 163)(61 164)(62 165)(63 166)(64 167)(65 168)(66 169)(67 170)(68 171)(69 172)(70 173)(71 174)(72 175)(73 176)(74 177)(75 178)(76 179)(77 180)(78 181)(79 182)(80 183)(81 184)(82 185)(83 186)(84 187)(85 188)(86 189)(87 190)(88 191)(89 192)(90 193)(91 194)(92 195)(93 196)(94 197)(95 198)(96 199)(97 200)(98 201)(99 202)(100 203)(101 204)(102 205)(103 206)(104 207)(105 208)(106 209)(107 210)(108 211)(109 212)(110 213)(111 214)(112 215)(113 216)(114 217)(115 218)(116 219)(117 220)(118 221)(119 222)(120 223)(121 224)(122 225)(123 226)(124 227)(125 228)(126 229)(127 230)(128 231)(129 232)(130 233)(131 234)(132 235)(133 236)(134 237)(135 238)(136 239)(137 240)(138 241)
(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)(139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276)
G:=sub<Sym(276)| (1,242)(2,243)(3,244)(4,245)(5,246)(6,247)(7,248)(8,249)(9,250)(10,251)(11,252)(12,253)(13,254)(14,255)(15,256)(16,257)(17,258)(18,259)(19,260)(20,261)(21,262)(22,263)(23,264)(24,265)(25,266)(26,267)(27,268)(28,269)(29,270)(30,271)(31,272)(32,273)(33,274)(34,275)(35,276)(36,139)(37,140)(38,141)(39,142)(40,143)(41,144)(42,145)(43,146)(44,147)(45,148)(46,149)(47,150)(48,151)(49,152)(50,153)(51,154)(52,155)(53,156)(54,157)(55,158)(56,159)(57,160)(58,161)(59,162)(60,163)(61,164)(62,165)(63,166)(64,167)(65,168)(66,169)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,176)(74,177)(75,178)(76,179)(77,180)(78,181)(79,182)(80,183)(81,184)(82,185)(83,186)(84,187)(85,188)(86,189)(87,190)(88,191)(89,192)(90,193)(91,194)(92,195)(93,196)(94,197)(95,198)(96,199)(97,200)(98,201)(99,202)(100,203)(101,204)(102,205)(103,206)(104,207)(105,208)(106,209)(107,210)(108,211)(109,212)(110,213)(111,214)(112,215)(113,216)(114,217)(115,218)(116,219)(117,220)(118,221)(119,222)(120,223)(121,224)(122,225)(123,226)(124,227)(125,228)(126,229)(127,230)(128,231)(129,232)(130,233)(131,234)(132,235)(133,236)(134,237)(135,238)(136,239)(137,240)(138,241), (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276)>;
G:=Group( (1,242)(2,243)(3,244)(4,245)(5,246)(6,247)(7,248)(8,249)(9,250)(10,251)(11,252)(12,253)(13,254)(14,255)(15,256)(16,257)(17,258)(18,259)(19,260)(20,261)(21,262)(22,263)(23,264)(24,265)(25,266)(26,267)(27,268)(28,269)(29,270)(30,271)(31,272)(32,273)(33,274)(34,275)(35,276)(36,139)(37,140)(38,141)(39,142)(40,143)(41,144)(42,145)(43,146)(44,147)(45,148)(46,149)(47,150)(48,151)(49,152)(50,153)(51,154)(52,155)(53,156)(54,157)(55,158)(56,159)(57,160)(58,161)(59,162)(60,163)(61,164)(62,165)(63,166)(64,167)(65,168)(66,169)(67,170)(68,171)(69,172)(70,173)(71,174)(72,175)(73,176)(74,177)(75,178)(76,179)(77,180)(78,181)(79,182)(80,183)(81,184)(82,185)(83,186)(84,187)(85,188)(86,189)(87,190)(88,191)(89,192)(90,193)(91,194)(92,195)(93,196)(94,197)(95,198)(96,199)(97,200)(98,201)(99,202)(100,203)(101,204)(102,205)(103,206)(104,207)(105,208)(106,209)(107,210)(108,211)(109,212)(110,213)(111,214)(112,215)(113,216)(114,217)(115,218)(116,219)(117,220)(118,221)(119,222)(120,223)(121,224)(122,225)(123,226)(124,227)(125,228)(126,229)(127,230)(128,231)(129,232)(130,233)(131,234)(132,235)(133,236)(134,237)(135,238)(136,239)(137,240)(138,241), (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)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276) );
G=PermutationGroup([[(1,242),(2,243),(3,244),(4,245),(5,246),(6,247),(7,248),(8,249),(9,250),(10,251),(11,252),(12,253),(13,254),(14,255),(15,256),(16,257),(17,258),(18,259),(19,260),(20,261),(21,262),(22,263),(23,264),(24,265),(25,266),(26,267),(27,268),(28,269),(29,270),(30,271),(31,272),(32,273),(33,274),(34,275),(35,276),(36,139),(37,140),(38,141),(39,142),(40,143),(41,144),(42,145),(43,146),(44,147),(45,148),(46,149),(47,150),(48,151),(49,152),(50,153),(51,154),(52,155),(53,156),(54,157),(55,158),(56,159),(57,160),(58,161),(59,162),(60,163),(61,164),(62,165),(63,166),(64,167),(65,168),(66,169),(67,170),(68,171),(69,172),(70,173),(71,174),(72,175),(73,176),(74,177),(75,178),(76,179),(77,180),(78,181),(79,182),(80,183),(81,184),(82,185),(83,186),(84,187),(85,188),(86,189),(87,190),(88,191),(89,192),(90,193),(91,194),(92,195),(93,196),(94,197),(95,198),(96,199),(97,200),(98,201),(99,202),(100,203),(101,204),(102,205),(103,206),(104,207),(105,208),(106,209),(107,210),(108,211),(109,212),(110,213),(111,214),(112,215),(113,216),(114,217),(115,218),(116,219),(117,220),(118,221),(119,222),(120,223),(121,224),(122,225),(123,226),(124,227),(125,228),(126,229),(127,230),(128,231),(129,232),(130,233),(131,234),(132,235),(133,236),(134,237),(135,238),(136,239),(137,240),(138,241)], [(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),(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276)]])
276 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 23A | ··· | 23V | 46A | ··· | 46BN | 69A | ··· | 69AR | 138A | ··· | 138EB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 | 138 | ··· | 138 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
276 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C23 | C46 | C69 | C138 |
| kernel | C2×C138 | C138 | C2×C46 | C46 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 22 | 66 | 44 | 132 |
Matrix representation of C2×C138 ►in GL2(𝔽139) generated by
| 138 | 0 |
| 0 | 1 |
| 37 | 0 |
| 0 | 138 |
G:=sub<GL(2,GF(139))| [138,0,0,1],[37,0,0,138] >;
C2×C138 in GAP, Magma, Sage, TeX
C_2\times C_{138} % in TeX
G:=Group("C2xC138"); // GroupNames label
G:=SmallGroup(276,10);
// by ID
G=gap.SmallGroup(276,10);
# by ID
G:=PCGroup([4,-2,-2,-3,-23]);
// Polycyclic
G:=Group<a,b|a^2=b^138=1,a*b=b*a>;
// generators/relations
Export