direct product, abelian, monomial, 2-elementary
Aliases: C2×C144, SmallGroup(288,59)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C144 |
| C1 — C2×C144 |
| C1 — C2×C144 |
Generators and relations for C2×C144
G = < a,b | a2=b144=1, ab=ba >
Smallest permutation representation of C2×C144
►Regular action on 288 points
Generators in S288
(1 216)(2 217)(3 218)(4 219)(5 220)(6 221)(7 222)(8 223)(9 224)(10 225)(11 226)(12 227)(13 228)(14 229)(15 230)(16 231)(17 232)(18 233)(19 234)(20 235)(21 236)(22 237)(23 238)(24 239)(25 240)(26 241)(27 242)(28 243)(29 244)(30 245)(31 246)(32 247)(33 248)(34 249)(35 250)(36 251)(37 252)(38 253)(39 254)(40 255)(41 256)(42 257)(43 258)(44 259)(45 260)(46 261)(47 262)(48 263)(49 264)(50 265)(51 266)(52 267)(53 268)(54 269)(55 270)(56 271)(57 272)(58 273)(59 274)(60 275)(61 276)(62 277)(63 278)(64 279)(65 280)(66 281)(67 282)(68 283)(69 284)(70 285)(71 286)(72 287)(73 288)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 161)(91 162)(92 163)(93 164)(94 165)(95 166)(96 167)(97 168)(98 169)(99 170)(100 171)(101 172)(102 173)(103 174)(104 175)(105 176)(106 177)(107 178)(108 179)(109 180)(110 181)(111 182)(112 183)(113 184)(114 185)(115 186)(116 187)(117 188)(118 189)(119 190)(120 191)(121 192)(122 193)(123 194)(124 195)(125 196)(126 197)(127 198)(128 199)(129 200)(130 201)(131 202)(132 203)(133 204)(134 205)(135 206)(136 207)(137 208)(138 209)(139 210)(140 211)(141 212)(142 213)(143 214)(144 215)
(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)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288)
G:=sub<Sym(288)| (1,216)(2,217)(3,218)(4,219)(5,220)(6,221)(7,222)(8,223)(9,224)(10,225)(11,226)(12,227)(13,228)(14,229)(15,230)(16,231)(17,232)(18,233)(19,234)(20,235)(21,236)(22,237)(23,238)(24,239)(25,240)(26,241)(27,242)(28,243)(29,244)(30,245)(31,246)(32,247)(33,248)(34,249)(35,250)(36,251)(37,252)(38,253)(39,254)(40,255)(41,256)(42,257)(43,258)(44,259)(45,260)(46,261)(47,262)(48,263)(49,264)(50,265)(51,266)(52,267)(53,268)(54,269)(55,270)(56,271)(57,272)(58,273)(59,274)(60,275)(61,276)(62,277)(63,278)(64,279)(65,280)(66,281)(67,282)(68,283)(69,284)(70,285)(71,286)(72,287)(73,288)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,161)(91,162)(92,163)(93,164)(94,165)(95,166)(96,167)(97,168)(98,169)(99,170)(100,171)(101,172)(102,173)(103,174)(104,175)(105,176)(106,177)(107,178)(108,179)(109,180)(110,181)(111,182)(112,183)(113,184)(114,185)(115,186)(116,187)(117,188)(118,189)(119,190)(120,191)(121,192)(122,193)(123,194)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(133,204)(134,205)(135,206)(136,207)(137,208)(138,209)(139,210)(140,211)(141,212)(142,213)(143,214)(144,215), (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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288)>;
G:=Group( (1,216)(2,217)(3,218)(4,219)(5,220)(6,221)(7,222)(8,223)(9,224)(10,225)(11,226)(12,227)(13,228)(14,229)(15,230)(16,231)(17,232)(18,233)(19,234)(20,235)(21,236)(22,237)(23,238)(24,239)(25,240)(26,241)(27,242)(28,243)(29,244)(30,245)(31,246)(32,247)(33,248)(34,249)(35,250)(36,251)(37,252)(38,253)(39,254)(40,255)(41,256)(42,257)(43,258)(44,259)(45,260)(46,261)(47,262)(48,263)(49,264)(50,265)(51,266)(52,267)(53,268)(54,269)(55,270)(56,271)(57,272)(58,273)(59,274)(60,275)(61,276)(62,277)(63,278)(64,279)(65,280)(66,281)(67,282)(68,283)(69,284)(70,285)(71,286)(72,287)(73,288)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,161)(91,162)(92,163)(93,164)(94,165)(95,166)(96,167)(97,168)(98,169)(99,170)(100,171)(101,172)(102,173)(103,174)(104,175)(105,176)(106,177)(107,178)(108,179)(109,180)(110,181)(111,182)(112,183)(113,184)(114,185)(115,186)(116,187)(117,188)(118,189)(119,190)(120,191)(121,192)(122,193)(123,194)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(133,204)(134,205)(135,206)(136,207)(137,208)(138,209)(139,210)(140,211)(141,212)(142,213)(143,214)(144,215), (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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288) );
G=PermutationGroup([[(1,216),(2,217),(3,218),(4,219),(5,220),(6,221),(7,222),(8,223),(9,224),(10,225),(11,226),(12,227),(13,228),(14,229),(15,230),(16,231),(17,232),(18,233),(19,234),(20,235),(21,236),(22,237),(23,238),(24,239),(25,240),(26,241),(27,242),(28,243),(29,244),(30,245),(31,246),(32,247),(33,248),(34,249),(35,250),(36,251),(37,252),(38,253),(39,254),(40,255),(41,256),(42,257),(43,258),(44,259),(45,260),(46,261),(47,262),(48,263),(49,264),(50,265),(51,266),(52,267),(53,268),(54,269),(55,270),(56,271),(57,272),(58,273),(59,274),(60,275),(61,276),(62,277),(63,278),(64,279),(65,280),(66,281),(67,282),(68,283),(69,284),(70,285),(71,286),(72,287),(73,288),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151),(81,152),(82,153),(83,154),(84,155),(85,156),(86,157),(87,158),(88,159),(89,160),(90,161),(91,162),(92,163),(93,164),(94,165),(95,166),(96,167),(97,168),(98,169),(99,170),(100,171),(101,172),(102,173),(103,174),(104,175),(105,176),(106,177),(107,178),(108,179),(109,180),(110,181),(111,182),(112,183),(113,184),(114,185),(115,186),(116,187),(117,188),(118,189),(119,190),(120,191),(121,192),(122,193),(123,194),(124,195),(125,196),(126,197),(127,198),(128,199),(129,200),(130,201),(131,202),(132,203),(133,204),(134,205),(135,206),(136,207),(137,208),(138,209),(139,210),(140,211),(141,212),(142,213),(143,214),(144,215)], [(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),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288)]])
288 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 8A | ··· | 8H | 9A | ··· | 9F | 12A | ··· | 12H | 16A | ··· | 16P | 18A | ··· | 18R | 24A | ··· | 24P | 36A | ··· | 36X | 48A | ··· | 48AF | 72A | ··· | 72AV | 144A | ··· | 144CR |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 16 | ··· | 16 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 48 | ··· | 48 | 72 | ··· | 72 | 144 | ··· | 144 |
| 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 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
288 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C8 | C9 | C12 | C12 | C16 | C18 | C18 | C24 | C24 | C36 | C36 | C48 | C72 | C72 | C144 |
| kernel | C2×C144 | C144 | C2×C72 | C2×C48 | C72 | C2×C36 | C48 | C2×C24 | C36 | C2×C18 | C2×C16 | C24 | C2×C12 | C18 | C16 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 6 | 4 | 4 | 16 | 12 | 6 | 8 | 8 | 12 | 12 | 32 | 24 | 24 | 96 |
Matrix representation of C2×C144 ►in GL2(𝔽433) generated by
| 432 | 0 |
| 0 | 432 |
| 279 | 0 |
| 0 | 2 |
G:=sub<GL(2,GF(433))| [432,0,0,432],[279,0,0,2] >;
C2×C144 in GAP, Magma, Sage, TeX
C_2\times C_{144} % in TeX
G:=Group("C2xC144"); // GroupNames label
G:=SmallGroup(288,59);
// by ID
G=gap.SmallGroup(288,59);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-2,84,142,192,124]);
// Polycyclic
G:=Group<a,b|a^2=b^144=1,a*b=b*a>;
// generators/relations
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