direct product, abelian, monomial
Aliases: C2×C122, SmallGroup(288,811)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C122 |
| C1 — C2×C122 |
| C1 — C2×C122 |
Generators and relations for C2×C122
G = < a,b,c | a2=b12=c12=1, ab=ba, ac=ca, bc=cb >
Subgroups: 324, all normal (8 characteristic)
C1, C2, C3, C4, C22, C22, C6, C2×C4, C23, C32, C12, C2×C6, C42, C22×C4, C3×C6, C2×C12, C22×C6, C2×C42, C3×C12, C62, C62, C4×C12, C22×C12, C6×C12, C2×C62, C2×C4×C12, C122, C2×C6×C12, C2×C122
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C32, C12, C2×C6, C42, C22×C4, C3×C6, C2×C12, C22×C6, C2×C42, C3×C12, C62, C4×C12, C22×C12, C6×C12, C2×C62, C2×C4×C12, C122, C2×C6×C12, C2×C122
►Regular action on 288 points
Generators in S288
(1 253)(2 254)(3 255)(4 256)(5 257)(6 258)(7 259)(8 260)(9 261)(10 262)(11 263)(12 264)(13 228)(14 217)(15 218)(16 219)(17 220)(18 221)(19 222)(20 223)(21 224)(22 225)(23 226)(24 227)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 73)(34 74)(35 75)(36 76)(37 201)(38 202)(39 203)(40 204)(41 193)(42 194)(43 195)(44 196)(45 197)(46 198)(47 199)(48 200)(49 106)(50 107)(51 108)(52 97)(53 98)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 105)(61 128)(62 129)(63 130)(64 131)(65 132)(66 121)(67 122)(68 123)(69 124)(70 125)(71 126)(72 127)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(109 148)(110 149)(111 150)(112 151)(113 152)(114 153)(115 154)(116 155)(117 156)(118 145)(119 146)(120 147)(157 287)(158 288)(159 277)(160 278)(161 279)(162 280)(163 281)(164 282)(165 283)(166 284)(167 285)(168 286)(169 216)(170 205)(171 206)(172 207)(173 208)(174 209)(175 210)(176 211)(177 212)(178 213)(179 214)(180 215)(181 240)(182 229)(183 230)(184 231)(185 232)(186 233)(187 234)(188 235)(189 236)(190 237)(191 238)(192 239)(241 275)(242 276)(243 265)(244 266)(245 267)(246 268)(247 269)(248 270)(249 271)(250 272)(251 273)(252 274)
(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)
(1 59 203 136 173 126 113 244 31 288 188 18)(2 60 204 137 174 127 114 245 32 277 189 19)(3 49 193 138 175 128 115 246 33 278 190 20)(4 50 194 139 176 129 116 247 34 279 191 21)(5 51 195 140 177 130 117 248 35 280 192 22)(6 52 196 141 178 131 118 249 36 281 181 23)(7 53 197 142 179 132 119 250 25 282 182 24)(8 54 198 143 180 121 120 251 26 283 183 13)(9 55 199 144 169 122 109 252 27 284 184 14)(10 56 200 133 170 123 110 241 28 285 185 15)(11 57 201 134 171 124 111 242 29 286 186 16)(12 58 202 135 172 125 112 243 30 287 187 17)(37 86 206 69 150 276 81 168 233 219 263 102)(38 87 207 70 151 265 82 157 234 220 264 103)(39 88 208 71 152 266 83 158 235 221 253 104)(40 89 209 72 153 267 84 159 236 222 254 105)(41 90 210 61 154 268 73 160 237 223 255 106)(42 91 211 62 155 269 74 161 238 224 256 107)(43 92 212 63 156 270 75 162 239 225 257 108)(44 93 213 64 145 271 76 163 240 226 258 97)(45 94 214 65 146 272 77 164 229 227 259 98)(46 95 215 66 147 273 78 165 230 228 260 99)(47 96 216 67 148 274 79 166 231 217 261 100)(48 85 205 68 149 275 80 167 232 218 262 101)
G:=sub<Sym(288)| (1,253)(2,254)(3,255)(4,256)(5,257)(6,258)(7,259)(8,260)(9,261)(10,262)(11,263)(12,264)(13,228)(14,217)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(22,225)(23,226)(24,227)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,73)(34,74)(35,75)(36,76)(37,201)(38,202)(39,203)(40,204)(41,193)(42,194)(43,195)(44,196)(45,197)(46,198)(47,199)(48,200)(49,106)(50,107)(51,108)(52,97)(53,98)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(61,128)(62,129)(63,130)(64,131)(65,132)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,145)(119,146)(120,147)(157,287)(158,288)(159,277)(160,278)(161,279)(162,280)(163,281)(164,282)(165,283)(166,284)(167,285)(168,286)(169,216)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,211)(177,212)(178,213)(179,214)(180,215)(181,240)(182,229)(183,230)(184,231)(185,232)(186,233)(187,234)(188,235)(189,236)(190,237)(191,238)(192,239)(241,275)(242,276)(243,265)(244,266)(245,267)(246,268)(247,269)(248,270)(249,271)(250,272)(251,273)(252,274), (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), (1,59,203,136,173,126,113,244,31,288,188,18)(2,60,204,137,174,127,114,245,32,277,189,19)(3,49,193,138,175,128,115,246,33,278,190,20)(4,50,194,139,176,129,116,247,34,279,191,21)(5,51,195,140,177,130,117,248,35,280,192,22)(6,52,196,141,178,131,118,249,36,281,181,23)(7,53,197,142,179,132,119,250,25,282,182,24)(8,54,198,143,180,121,120,251,26,283,183,13)(9,55,199,144,169,122,109,252,27,284,184,14)(10,56,200,133,170,123,110,241,28,285,185,15)(11,57,201,134,171,124,111,242,29,286,186,16)(12,58,202,135,172,125,112,243,30,287,187,17)(37,86,206,69,150,276,81,168,233,219,263,102)(38,87,207,70,151,265,82,157,234,220,264,103)(39,88,208,71,152,266,83,158,235,221,253,104)(40,89,209,72,153,267,84,159,236,222,254,105)(41,90,210,61,154,268,73,160,237,223,255,106)(42,91,211,62,155,269,74,161,238,224,256,107)(43,92,212,63,156,270,75,162,239,225,257,108)(44,93,213,64,145,271,76,163,240,226,258,97)(45,94,214,65,146,272,77,164,229,227,259,98)(46,95,215,66,147,273,78,165,230,228,260,99)(47,96,216,67,148,274,79,166,231,217,261,100)(48,85,205,68,149,275,80,167,232,218,262,101)>;
G:=Group( (1,253)(2,254)(3,255)(4,256)(5,257)(6,258)(7,259)(8,260)(9,261)(10,262)(11,263)(12,264)(13,228)(14,217)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(22,225)(23,226)(24,227)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,73)(34,74)(35,75)(36,76)(37,201)(38,202)(39,203)(40,204)(41,193)(42,194)(43,195)(44,196)(45,197)(46,198)(47,199)(48,200)(49,106)(50,107)(51,108)(52,97)(53,98)(54,99)(55,100)(56,101)(57,102)(58,103)(59,104)(60,105)(61,128)(62,129)(63,130)(64,131)(65,132)(66,121)(67,122)(68,123)(69,124)(70,125)(71,126)(72,127)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(109,148)(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(118,145)(119,146)(120,147)(157,287)(158,288)(159,277)(160,278)(161,279)(162,280)(163,281)(164,282)(165,283)(166,284)(167,285)(168,286)(169,216)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,211)(177,212)(178,213)(179,214)(180,215)(181,240)(182,229)(183,230)(184,231)(185,232)(186,233)(187,234)(188,235)(189,236)(190,237)(191,238)(192,239)(241,275)(242,276)(243,265)(244,266)(245,267)(246,268)(247,269)(248,270)(249,271)(250,272)(251,273)(252,274), (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), (1,59,203,136,173,126,113,244,31,288,188,18)(2,60,204,137,174,127,114,245,32,277,189,19)(3,49,193,138,175,128,115,246,33,278,190,20)(4,50,194,139,176,129,116,247,34,279,191,21)(5,51,195,140,177,130,117,248,35,280,192,22)(6,52,196,141,178,131,118,249,36,281,181,23)(7,53,197,142,179,132,119,250,25,282,182,24)(8,54,198,143,180,121,120,251,26,283,183,13)(9,55,199,144,169,122,109,252,27,284,184,14)(10,56,200,133,170,123,110,241,28,285,185,15)(11,57,201,134,171,124,111,242,29,286,186,16)(12,58,202,135,172,125,112,243,30,287,187,17)(37,86,206,69,150,276,81,168,233,219,263,102)(38,87,207,70,151,265,82,157,234,220,264,103)(39,88,208,71,152,266,83,158,235,221,253,104)(40,89,209,72,153,267,84,159,236,222,254,105)(41,90,210,61,154,268,73,160,237,223,255,106)(42,91,211,62,155,269,74,161,238,224,256,107)(43,92,212,63,156,270,75,162,239,225,257,108)(44,93,213,64,145,271,76,163,240,226,258,97)(45,94,214,65,146,272,77,164,229,227,259,98)(46,95,215,66,147,273,78,165,230,228,260,99)(47,96,216,67,148,274,79,166,231,217,261,100)(48,85,205,68,149,275,80,167,232,218,262,101) );
G=PermutationGroup([[(1,253),(2,254),(3,255),(4,256),(5,257),(6,258),(7,259),(8,260),(9,261),(10,262),(11,263),(12,264),(13,228),(14,217),(15,218),(16,219),(17,220),(18,221),(19,222),(20,223),(21,224),(22,225),(23,226),(24,227),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,73),(34,74),(35,75),(36,76),(37,201),(38,202),(39,203),(40,204),(41,193),(42,194),(43,195),(44,196),(45,197),(46,198),(47,199),(48,200),(49,106),(50,107),(51,108),(52,97),(53,98),(54,99),(55,100),(56,101),(57,102),(58,103),(59,104),(60,105),(61,128),(62,129),(63,130),(64,131),(65,132),(66,121),(67,122),(68,123),(69,124),(70,125),(71,126),(72,127),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,141),(94,142),(95,143),(96,144),(109,148),(110,149),(111,150),(112,151),(113,152),(114,153),(115,154),(116,155),(117,156),(118,145),(119,146),(120,147),(157,287),(158,288),(159,277),(160,278),(161,279),(162,280),(163,281),(164,282),(165,283),(166,284),(167,285),(168,286),(169,216),(170,205),(171,206),(172,207),(173,208),(174,209),(175,210),(176,211),(177,212),(178,213),(179,214),(180,215),(181,240),(182,229),(183,230),(184,231),(185,232),(186,233),(187,234),(188,235),(189,236),(190,237),(191,238),(192,239),(241,275),(242,276),(243,265),(244,266),(245,267),(246,268),(247,269),(248,270),(249,271),(250,272),(251,273),(252,274)], [(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)], [(1,59,203,136,173,126,113,244,31,288,188,18),(2,60,204,137,174,127,114,245,32,277,189,19),(3,49,193,138,175,128,115,246,33,278,190,20),(4,50,194,139,176,129,116,247,34,279,191,21),(5,51,195,140,177,130,117,248,35,280,192,22),(6,52,196,141,178,131,118,249,36,281,181,23),(7,53,197,142,179,132,119,250,25,282,182,24),(8,54,198,143,180,121,120,251,26,283,183,13),(9,55,199,144,169,122,109,252,27,284,184,14),(10,56,200,133,170,123,110,241,28,285,185,15),(11,57,201,134,171,124,111,242,29,286,186,16),(12,58,202,135,172,125,112,243,30,287,187,17),(37,86,206,69,150,276,81,168,233,219,263,102),(38,87,207,70,151,265,82,157,234,220,264,103),(39,88,208,71,152,266,83,158,235,221,253,104),(40,89,209,72,153,267,84,159,236,222,254,105),(41,90,210,61,154,268,73,160,237,223,255,106),(42,91,211,62,155,269,74,161,238,224,256,107),(43,92,212,63,156,270,75,162,239,225,257,108),(44,93,213,64,145,271,76,163,240,226,258,97),(45,94,214,65,146,272,77,164,229,227,259,98),(46,95,215,66,147,273,78,165,230,228,260,99),(47,96,216,67,148,274,79,166,231,217,261,100),(48,85,205,68,149,275,80,167,232,218,262,101)]])
288 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | ··· | 3H | 4A | ··· | 4X | 6A | ··· | 6BD | 12A | ··· | 12GJ |
| order | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
288 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 |
| kernel | C2×C122 | C122 | C2×C6×C12 | C2×C4×C12 | C6×C12 | C4×C12 | C22×C12 | C2×C12 |
| # reps | 1 | 4 | 3 | 8 | 24 | 32 | 24 | 192 |
Matrix representation of C2×C122 ►in GL4(𝔽13) generated by
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,1],[8,0,0,0,0,1,0,0,0,0,6,0,0,0,0,8],[12,0,0,0,0,10,0,0,0,0,7,0,0,0,0,1] >;
C2×C122 in GAP, Magma, Sage, TeX
C_2\times C_{12}^2 % in TeX
G:=Group("C2xC12^2"); // GroupNames label
G:=SmallGroup(288,811);
// by ID
G=gap.SmallGroup(288,811);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,504,1016]);
// Polycyclic
G:=Group<a,b,c|a^2=b^12=c^12=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations