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## G = C2×C122order 288 = 25·32

### Abelian group of type [2,12,12]

Aliases: C2×C122, SmallGroup(288,811)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C122
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C6×C12 — C122 — C2×C122
 Lower central C1 — C2×C122
 Upper central 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 [×7], C3 [×4], C4 [×12], C22, C22 [×6], C6 [×28], C2×C4 [×18], C23, C32, C12 [×48], C2×C6 [×28], C42 [×4], C22×C4 [×3], C3×C6 [×7], C2×C12 [×72], C22×C6 [×4], C2×C42, C3×C12 [×12], C62, C62 [×6], C4×C12 [×16], C22×C12 [×12], C6×C12 [×18], C2×C62, C2×C4×C12 [×4], C122 [×4], C2×C6×C12 [×3], C2×C122
Quotients: C1, C2 [×7], C3 [×4], C4 [×12], C22 [×7], C6 [×28], C2×C4 [×18], C23, C32, C12 [×48], C2×C6 [×28], C42 [×4], C22×C4 [×3], C3×C6 [×7], C2×C12 [×72], C22×C6 [×4], C2×C42, C3×C12 [×12], C62 [×7], C4×C12 [×16], C22×C12 [×12], C6×C12 [×18], C2×C62, C2×C4×C12 [×4], C122 [×4], C2×C6×C12 [×3], C2×C122

Smallest permutation representation of C2×C122
Regular action on 288 points
Generators in S288
(1 147)(2 148)(3 149)(4 150)(5 151)(6 152)(7 153)(8 154)(9 155)(10 156)(11 145)(12 146)(13 132)(14 121)(15 122)(16 123)(17 124)(18 125)(19 126)(20 127)(21 128)(22 129)(23 130)(24 131)(25 203)(26 204)(27 193)(28 194)(29 195)(30 196)(31 197)(32 198)(33 199)(34 200)(35 201)(36 202)(37 252)(38 241)(39 242)(40 243)(41 244)(42 245)(43 246)(44 247)(45 248)(46 249)(47 250)(48 251)(49 222)(50 223)(51 224)(52 225)(53 226)(54 227)(55 228)(56 217)(57 218)(58 219)(59 220)(60 221)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 141)(68 142)(69 143)(70 144)(71 133)(72 134)(73 164)(74 165)(75 166)(76 167)(77 168)(78 157)(79 158)(80 159)(81 160)(82 161)(83 162)(84 163)(85 263)(86 264)(87 253)(88 254)(89 255)(90 256)(91 257)(92 258)(93 259)(94 260)(95 261)(96 262)(97 216)(98 205)(99 206)(100 207)(101 208)(102 209)(103 210)(104 211)(105 212)(106 213)(107 214)(108 215)(109 185)(110 186)(111 187)(112 188)(113 189)(114 190)(115 191)(116 192)(117 181)(118 182)(119 183)(120 184)(169 281)(170 282)(171 283)(172 284)(173 285)(174 286)(175 287)(176 288)(177 277)(178 278)(179 279)(180 280)(229 271)(230 272)(231 273)(232 274)(233 275)(234 276)(235 265)(236 266)(237 267)(238 268)(239 269)(240 270)
(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)
(1 252 203 226 276 69 187 104 20 261 280 164)(2 241 204 227 265 70 188 105 21 262 281 165)(3 242 193 228 266 71 189 106 22 263 282 166)(4 243 194 217 267 72 190 107 23 264 283 167)(5 244 195 218 268 61 191 108 24 253 284 168)(6 245 196 219 269 62 192 97 13 254 285 157)(7 246 197 220 270 63 181 98 14 255 286 158)(8 247 198 221 271 64 182 99 15 256 287 159)(9 248 199 222 272 65 183 100 16 257 288 160)(10 249 200 223 273 66 184 101 17 258 277 161)(11 250 201 224 274 67 185 102 18 259 278 162)(12 251 202 225 275 68 186 103 19 260 279 163)(25 53 234 143 111 211 127 95 180 73 147 37)(26 54 235 144 112 212 128 96 169 74 148 38)(27 55 236 133 113 213 129 85 170 75 149 39)(28 56 237 134 114 214 130 86 171 76 150 40)(29 57 238 135 115 215 131 87 172 77 151 41)(30 58 239 136 116 216 132 88 173 78 152 42)(31 59 240 137 117 205 121 89 174 79 153 43)(32 60 229 138 118 206 122 90 175 80 154 44)(33 49 230 139 119 207 123 91 176 81 155 45)(34 50 231 140 120 208 124 92 177 82 156 46)(35 51 232 141 109 209 125 93 178 83 145 47)(36 52 233 142 110 210 126 94 179 84 146 48)

G:=sub<Sym(288)| (1,147)(2,148)(3,149)(4,150)(5,151)(6,152)(7,153)(8,154)(9,155)(10,156)(11,145)(12,146)(13,132)(14,121)(15,122)(16,123)(17,124)(18,125)(19,126)(20,127)(21,128)(22,129)(23,130)(24,131)(25,203)(26,204)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,252)(38,241)(39,242)(40,243)(41,244)(42,245)(43,246)(44,247)(45,248)(46,249)(47,250)(48,251)(49,222)(50,223)(51,224)(52,225)(53,226)(54,227)(55,228)(56,217)(57,218)(58,219)(59,220)(60,221)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,133)(72,134)(73,164)(74,165)(75,166)(76,167)(77,168)(78,157)(79,158)(80,159)(81,160)(82,161)(83,162)(84,163)(85,263)(86,264)(87,253)(88,254)(89,255)(90,256)(91,257)(92,258)(93,259)(94,260)(95,261)(96,262)(97,216)(98,205)(99,206)(100,207)(101,208)(102,209)(103,210)(104,211)(105,212)(106,213)(107,214)(108,215)(109,185)(110,186)(111,187)(112,188)(113,189)(114,190)(115,191)(116,192)(117,181)(118,182)(119,183)(120,184)(169,281)(170,282)(171,283)(172,284)(173,285)(174,286)(175,287)(176,288)(177,277)(178,278)(179,279)(180,280)(229,271)(230,272)(231,273)(232,274)(233,275)(234,276)(235,265)(236,266)(237,267)(238,268)(239,269)(240,270), (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), (1,252,203,226,276,69,187,104,20,261,280,164)(2,241,204,227,265,70,188,105,21,262,281,165)(3,242,193,228,266,71,189,106,22,263,282,166)(4,243,194,217,267,72,190,107,23,264,283,167)(5,244,195,218,268,61,191,108,24,253,284,168)(6,245,196,219,269,62,192,97,13,254,285,157)(7,246,197,220,270,63,181,98,14,255,286,158)(8,247,198,221,271,64,182,99,15,256,287,159)(9,248,199,222,272,65,183,100,16,257,288,160)(10,249,200,223,273,66,184,101,17,258,277,161)(11,250,201,224,274,67,185,102,18,259,278,162)(12,251,202,225,275,68,186,103,19,260,279,163)(25,53,234,143,111,211,127,95,180,73,147,37)(26,54,235,144,112,212,128,96,169,74,148,38)(27,55,236,133,113,213,129,85,170,75,149,39)(28,56,237,134,114,214,130,86,171,76,150,40)(29,57,238,135,115,215,131,87,172,77,151,41)(30,58,239,136,116,216,132,88,173,78,152,42)(31,59,240,137,117,205,121,89,174,79,153,43)(32,60,229,138,118,206,122,90,175,80,154,44)(33,49,230,139,119,207,123,91,176,81,155,45)(34,50,231,140,120,208,124,92,177,82,156,46)(35,51,232,141,109,209,125,93,178,83,145,47)(36,52,233,142,110,210,126,94,179,84,146,48)>;

G:=Group( (1,147)(2,148)(3,149)(4,150)(5,151)(6,152)(7,153)(8,154)(9,155)(10,156)(11,145)(12,146)(13,132)(14,121)(15,122)(16,123)(17,124)(18,125)(19,126)(20,127)(21,128)(22,129)(23,130)(24,131)(25,203)(26,204)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,252)(38,241)(39,242)(40,243)(41,244)(42,245)(43,246)(44,247)(45,248)(46,249)(47,250)(48,251)(49,222)(50,223)(51,224)(52,225)(53,226)(54,227)(55,228)(56,217)(57,218)(58,219)(59,220)(60,221)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,133)(72,134)(73,164)(74,165)(75,166)(76,167)(77,168)(78,157)(79,158)(80,159)(81,160)(82,161)(83,162)(84,163)(85,263)(86,264)(87,253)(88,254)(89,255)(90,256)(91,257)(92,258)(93,259)(94,260)(95,261)(96,262)(97,216)(98,205)(99,206)(100,207)(101,208)(102,209)(103,210)(104,211)(105,212)(106,213)(107,214)(108,215)(109,185)(110,186)(111,187)(112,188)(113,189)(114,190)(115,191)(116,192)(117,181)(118,182)(119,183)(120,184)(169,281)(170,282)(171,283)(172,284)(173,285)(174,286)(175,287)(176,288)(177,277)(178,278)(179,279)(180,280)(229,271)(230,272)(231,273)(232,274)(233,275)(234,276)(235,265)(236,266)(237,267)(238,268)(239,269)(240,270), (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), (1,252,203,226,276,69,187,104,20,261,280,164)(2,241,204,227,265,70,188,105,21,262,281,165)(3,242,193,228,266,71,189,106,22,263,282,166)(4,243,194,217,267,72,190,107,23,264,283,167)(5,244,195,218,268,61,191,108,24,253,284,168)(6,245,196,219,269,62,192,97,13,254,285,157)(7,246,197,220,270,63,181,98,14,255,286,158)(8,247,198,221,271,64,182,99,15,256,287,159)(9,248,199,222,272,65,183,100,16,257,288,160)(10,249,200,223,273,66,184,101,17,258,277,161)(11,250,201,224,274,67,185,102,18,259,278,162)(12,251,202,225,275,68,186,103,19,260,279,163)(25,53,234,143,111,211,127,95,180,73,147,37)(26,54,235,144,112,212,128,96,169,74,148,38)(27,55,236,133,113,213,129,85,170,75,149,39)(28,56,237,134,114,214,130,86,171,76,150,40)(29,57,238,135,115,215,131,87,172,77,151,41)(30,58,239,136,116,216,132,88,173,78,152,42)(31,59,240,137,117,205,121,89,174,79,153,43)(32,60,229,138,118,206,122,90,175,80,154,44)(33,49,230,139,119,207,123,91,176,81,155,45)(34,50,231,140,120,208,124,92,177,82,156,46)(35,51,232,141,109,209,125,93,178,83,145,47)(36,52,233,142,110,210,126,94,179,84,146,48) );

G=PermutationGroup([(1,147),(2,148),(3,149),(4,150),(5,151),(6,152),(7,153),(8,154),(9,155),(10,156),(11,145),(12,146),(13,132),(14,121),(15,122),(16,123),(17,124),(18,125),(19,126),(20,127),(21,128),(22,129),(23,130),(24,131),(25,203),(26,204),(27,193),(28,194),(29,195),(30,196),(31,197),(32,198),(33,199),(34,200),(35,201),(36,202),(37,252),(38,241),(39,242),(40,243),(41,244),(42,245),(43,246),(44,247),(45,248),(46,249),(47,250),(48,251),(49,222),(50,223),(51,224),(52,225),(53,226),(54,227),(55,228),(56,217),(57,218),(58,219),(59,220),(60,221),(61,135),(62,136),(63,137),(64,138),(65,139),(66,140),(67,141),(68,142),(69,143),(70,144),(71,133),(72,134),(73,164),(74,165),(75,166),(76,167),(77,168),(78,157),(79,158),(80,159),(81,160),(82,161),(83,162),(84,163),(85,263),(86,264),(87,253),(88,254),(89,255),(90,256),(91,257),(92,258),(93,259),(94,260),(95,261),(96,262),(97,216),(98,205),(99,206),(100,207),(101,208),(102,209),(103,210),(104,211),(105,212),(106,213),(107,214),(108,215),(109,185),(110,186),(111,187),(112,188),(113,189),(114,190),(115,191),(116,192),(117,181),(118,182),(119,183),(120,184),(169,281),(170,282),(171,283),(172,284),(173,285),(174,286),(175,287),(176,288),(177,277),(178,278),(179,279),(180,280),(229,271),(230,272),(231,273),(232,274),(233,275),(234,276),(235,265),(236,266),(237,267),(238,268),(239,269),(240,270)], [(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)], [(1,252,203,226,276,69,187,104,20,261,280,164),(2,241,204,227,265,70,188,105,21,262,281,165),(3,242,193,228,266,71,189,106,22,263,282,166),(4,243,194,217,267,72,190,107,23,264,283,167),(5,244,195,218,268,61,191,108,24,253,284,168),(6,245,196,219,269,62,192,97,13,254,285,157),(7,246,197,220,270,63,181,98,14,255,286,158),(8,247,198,221,271,64,182,99,15,256,287,159),(9,248,199,222,272,65,183,100,16,257,288,160),(10,249,200,223,273,66,184,101,17,258,277,161),(11,250,201,224,274,67,185,102,18,259,278,162),(12,251,202,225,275,68,186,103,19,260,279,163),(25,53,234,143,111,211,127,95,180,73,147,37),(26,54,235,144,112,212,128,96,169,74,148,38),(27,55,236,133,113,213,129,85,170,75,149,39),(28,56,237,134,114,214,130,86,171,76,150,40),(29,57,238,135,115,215,131,87,172,77,151,41),(30,58,239,136,116,216,132,88,173,78,152,42),(31,59,240,137,117,205,121,89,174,79,153,43),(32,60,229,138,118,206,122,90,175,80,154,44),(33,49,230,139,119,207,123,91,176,81,155,45),(34,50,231,140,120,208,124,92,177,82,156,46),(35,51,232,141,109,209,125,93,178,83,145,47),(36,52,233,142,110,210,126,94,179,84,146,48)])

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

׿
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