direct product, abelian, monomial, 2-elementary
Aliases: C2×C136, SmallGroup(272,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C136 |
| C1 — C2×C136 |
| C1 — C2×C136 |
Generators and relations for C2×C136
G = < a,b | a2=b136=1, ab=ba >
Smallest permutation representation of C2×C136
►Regular action on 272 points
Generators in S272
(1 206)(2 207)(3 208)(4 209)(5 210)(6 211)(7 212)(8 213)(9 214)(10 215)(11 216)(12 217)(13 218)(14 219)(15 220)(16 221)(17 222)(18 223)(19 224)(20 225)(21 226)(22 227)(23 228)(24 229)(25 230)(26 231)(27 232)(28 233)(29 234)(30 235)(31 236)(32 237)(33 238)(34 239)(35 240)(36 241)(37 242)(38 243)(39 244)(40 245)(41 246)(42 247)(43 248)(44 249)(45 250)(46 251)(47 252)(48 253)(49 254)(50 255)(51 256)(52 257)(53 258)(54 259)(55 260)(56 261)(57 262)(58 263)(59 264)(60 265)(61 266)(62 267)(63 268)(64 269)(65 270)(66 271)(67 272)(68 137)(69 138)(70 139)(71 140)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(81 150)(82 151)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)(91 160)(92 161)(93 162)(94 163)(95 164)(96 165)(97 166)(98 167)(99 168)(100 169)(101 170)(102 171)(103 172)(104 173)(105 174)(106 175)(107 176)(108 177)(109 178)(110 179)(111 180)(112 181)(113 182)(114 183)(115 184)(116 185)(117 186)(118 187)(119 188)(120 189)(121 190)(122 191)(123 192)(124 193)(125 194)(126 195)(127 196)(128 197)(129 198)(130 199)(131 200)(132 201)(133 202)(134 203)(135 204)(136 205)
(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)
G:=sub<Sym(272)| (1,206)(2,207)(3,208)(4,209)(5,210)(6,211)(7,212)(8,213)(9,214)(10,215)(11,216)(12,217)(13,218)(14,219)(15,220)(16,221)(17,222)(18,223)(19,224)(20,225)(21,226)(22,227)(23,228)(24,229)(25,230)(26,231)(27,232)(28,233)(29,234)(30,235)(31,236)(32,237)(33,238)(34,239)(35,240)(36,241)(37,242)(38,243)(39,244)(40,245)(41,246)(42,247)(43,248)(44,249)(45,250)(46,251)(47,252)(48,253)(49,254)(50,255)(51,256)(52,257)(53,258)(54,259)(55,260)(56,261)(57,262)(58,263)(59,264)(60,265)(61,266)(62,267)(63,268)(64,269)(65,270)(66,271)(67,272)(68,137)(69,138)(70,139)(71,140)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,161)(93,162)(94,163)(95,164)(96,165)(97,166)(98,167)(99,168)(100,169)(101,170)(102,171)(103,172)(104,173)(105,174)(106,175)(107,176)(108,177)(109,178)(110,179)(111,180)(112,181)(113,182)(114,183)(115,184)(116,185)(117,186)(118,187)(119,188)(120,189)(121,190)(122,191)(123,192)(124,193)(125,194)(126,195)(127,196)(128,197)(129,198)(130,199)(131,200)(132,201)(133,202)(134,203)(135,204)(136,205), (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)>;
G:=Group( (1,206)(2,207)(3,208)(4,209)(5,210)(6,211)(7,212)(8,213)(9,214)(10,215)(11,216)(12,217)(13,218)(14,219)(15,220)(16,221)(17,222)(18,223)(19,224)(20,225)(21,226)(22,227)(23,228)(24,229)(25,230)(26,231)(27,232)(28,233)(29,234)(30,235)(31,236)(32,237)(33,238)(34,239)(35,240)(36,241)(37,242)(38,243)(39,244)(40,245)(41,246)(42,247)(43,248)(44,249)(45,250)(46,251)(47,252)(48,253)(49,254)(50,255)(51,256)(52,257)(53,258)(54,259)(55,260)(56,261)(57,262)(58,263)(59,264)(60,265)(61,266)(62,267)(63,268)(64,269)(65,270)(66,271)(67,272)(68,137)(69,138)(70,139)(71,140)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,161)(93,162)(94,163)(95,164)(96,165)(97,166)(98,167)(99,168)(100,169)(101,170)(102,171)(103,172)(104,173)(105,174)(106,175)(107,176)(108,177)(109,178)(110,179)(111,180)(112,181)(113,182)(114,183)(115,184)(116,185)(117,186)(118,187)(119,188)(120,189)(121,190)(122,191)(123,192)(124,193)(125,194)(126,195)(127,196)(128,197)(129,198)(130,199)(131,200)(132,201)(133,202)(134,203)(135,204)(136,205), (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) );
G=PermutationGroup([[(1,206),(2,207),(3,208),(4,209),(5,210),(6,211),(7,212),(8,213),(9,214),(10,215),(11,216),(12,217),(13,218),(14,219),(15,220),(16,221),(17,222),(18,223),(19,224),(20,225),(21,226),(22,227),(23,228),(24,229),(25,230),(26,231),(27,232),(28,233),(29,234),(30,235),(31,236),(32,237),(33,238),(34,239),(35,240),(36,241),(37,242),(38,243),(39,244),(40,245),(41,246),(42,247),(43,248),(44,249),(45,250),(46,251),(47,252),(48,253),(49,254),(50,255),(51,256),(52,257),(53,258),(54,259),(55,260),(56,261),(57,262),(58,263),(59,264),(60,265),(61,266),(62,267),(63,268),(64,269),(65,270),(66,271),(67,272),(68,137),(69,138),(70,139),(71,140),(72,141),(73,142),(74,143),(75,144),(76,145),(77,146),(78,147),(79,148),(80,149),(81,150),(82,151),(83,152),(84,153),(85,154),(86,155),(87,156),(88,157),(89,158),(90,159),(91,160),(92,161),(93,162),(94,163),(95,164),(96,165),(97,166),(98,167),(99,168),(100,169),(101,170),(102,171),(103,172),(104,173),(105,174),(106,175),(107,176),(108,177),(109,178),(110,179),(111,180),(112,181),(113,182),(114,183),(115,184),(116,185),(117,186),(118,187),(119,188),(120,189),(121,190),(122,191),(123,192),(124,193),(125,194),(126,195),(127,196),(128,197),(129,198),(130,199),(131,200),(132,201),(133,202),(134,203),(135,204),(136,205)], [(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)]])
272 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 17A | ··· | 17P | 34A | ··· | 34AV | 68A | ··· | 68BL | 136A | ··· | 136DX |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 17 | ··· | 17 | 34 | ··· | 34 | 68 | ··· | 68 | 136 | ··· | 136 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
272 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C17 | C34 | C34 | C68 | C68 | C136 |
| kernel | C2×C136 | C136 | C2×C68 | C68 | C2×C34 | C34 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 16 | 32 | 16 | 32 | 32 | 128 |
Matrix representation of C2×C136 ►in GL2(𝔽137) generated by
| 1 | 0 |
| 0 | 136 |
| 116 | 0 |
| 0 | 53 |
G:=sub<GL(2,GF(137))| [1,0,0,136],[116,0,0,53] >;
C2×C136 in GAP, Magma, Sage, TeX
C_2\times C_{136} % in TeX
G:=Group("C2xC136"); // GroupNames label
G:=SmallGroup(272,23);
// by ID
G=gap.SmallGroup(272,23);
# by ID
G:=PCGroup([5,-2,-2,-17,-2,-2,340,58]);
// Polycyclic
G:=Group<a,b|a^2=b^136=1,a*b=b*a>;
// generators/relations
Export