direct product, abelian, monomial, 2-elementary
Aliases: C2×C188, SmallGroup(376,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C188 |
| C1 — C2×C188 |
| C1 — C2×C188 |
Generators and relations for C2×C188
G = < a,b | a2=b188=1, ab=ba >
Smallest permutation representation of C2×C188
►Regular action on 376 points
Generators in S376
(1 354)(2 355)(3 356)(4 357)(5 358)(6 359)(7 360)(8 361)(9 362)(10 363)(11 364)(12 365)(13 366)(14 367)(15 368)(16 369)(17 370)(18 371)(19 372)(20 373)(21 374)(22 375)(23 376)(24 189)(25 190)(26 191)(27 192)(28 193)(29 194)(30 195)(31 196)(32 197)(33 198)(34 199)(35 200)(36 201)(37 202)(38 203)(39 204)(40 205)(41 206)(42 207)(43 208)(44 209)(45 210)(46 211)(47 212)(48 213)(49 214)(50 215)(51 216)(52 217)(53 218)(54 219)(55 220)(56 221)(57 222)(58 223)(59 224)(60 225)(61 226)(62 227)(63 228)(64 229)(65 230)(66 231)(67 232)(68 233)(69 234)(70 235)(71 236)(72 237)(73 238)(74 239)(75 240)(76 241)(77 242)(78 243)(79 244)(80 245)(81 246)(82 247)(83 248)(84 249)(85 250)(86 251)(87 252)(88 253)(89 254)(90 255)(91 256)(92 257)(93 258)(94 259)(95 260)(96 261)(97 262)(98 263)(99 264)(100 265)(101 266)(102 267)(103 268)(104 269)(105 270)(106 271)(107 272)(108 273)(109 274)(110 275)(111 276)(112 277)(113 278)(114 279)(115 280)(116 281)(117 282)(118 283)(119 284)(120 285)(121 286)(122 287)(123 288)(124 289)(125 290)(126 291)(127 292)(128 293)(129 294)(130 295)(131 296)(132 297)(133 298)(134 299)(135 300)(136 301)(137 302)(138 303)(139 304)(140 305)(141 306)(142 307)(143 308)(144 309)(145 310)(146 311)(147 312)(148 313)(149 314)(150 315)(151 316)(152 317)(153 318)(154 319)(155 320)(156 321)(157 322)(158 323)(159 324)(160 325)(161 326)(162 327)(163 328)(164 329)(165 330)(166 331)(167 332)(168 333)(169 334)(170 335)(171 336)(172 337)(173 338)(174 339)(175 340)(176 341)(177 342)(178 343)(179 344)(180 345)(181 346)(182 347)(183 348)(184 349)(185 350)(186 351)(187 352)(188 353)
(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)(189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376)
G:=sub<Sym(376)| (1,354)(2,355)(3,356)(4,357)(5,358)(6,359)(7,360)(8,361)(9,362)(10,363)(11,364)(12,365)(13,366)(14,367)(15,368)(16,369)(17,370)(18,371)(19,372)(20,373)(21,374)(22,375)(23,376)(24,189)(25,190)(26,191)(27,192)(28,193)(29,194)(30,195)(31,196)(32,197)(33,198)(34,199)(35,200)(36,201)(37,202)(38,203)(39,204)(40,205)(41,206)(42,207)(43,208)(44,209)(45,210)(46,211)(47,212)(48,213)(49,214)(50,215)(51,216)(52,217)(53,218)(54,219)(55,220)(56,221)(57,222)(58,223)(59,224)(60,225)(61,226)(62,227)(63,228)(64,229)(65,230)(66,231)(67,232)(68,233)(69,234)(70,235)(71,236)(72,237)(73,238)(74,239)(75,240)(76,241)(77,242)(78,243)(79,244)(80,245)(81,246)(82,247)(83,248)(84,249)(85,250)(86,251)(87,252)(88,253)(89,254)(90,255)(91,256)(92,257)(93,258)(94,259)(95,260)(96,261)(97,262)(98,263)(99,264)(100,265)(101,266)(102,267)(103,268)(104,269)(105,270)(106,271)(107,272)(108,273)(109,274)(110,275)(111,276)(112,277)(113,278)(114,279)(115,280)(116,281)(117,282)(118,283)(119,284)(120,285)(121,286)(122,287)(123,288)(124,289)(125,290)(126,291)(127,292)(128,293)(129,294)(130,295)(131,296)(132,297)(133,298)(134,299)(135,300)(136,301)(137,302)(138,303)(139,304)(140,305)(141,306)(142,307)(143,308)(144,309)(145,310)(146,311)(147,312)(148,313)(149,314)(150,315)(151,316)(152,317)(153,318)(154,319)(155,320)(156,321)(157,322)(158,323)(159,324)(160,325)(161,326)(162,327)(163,328)(164,329)(165,330)(166,331)(167,332)(168,333)(169,334)(170,335)(171,336)(172,337)(173,338)(174,339)(175,340)(176,341)(177,342)(178,343)(179,344)(180,345)(181,346)(182,347)(183,348)(184,349)(185,350)(186,351)(187,352)(188,353), (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)(189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376)>;
G:=Group( (1,354)(2,355)(3,356)(4,357)(5,358)(6,359)(7,360)(8,361)(9,362)(10,363)(11,364)(12,365)(13,366)(14,367)(15,368)(16,369)(17,370)(18,371)(19,372)(20,373)(21,374)(22,375)(23,376)(24,189)(25,190)(26,191)(27,192)(28,193)(29,194)(30,195)(31,196)(32,197)(33,198)(34,199)(35,200)(36,201)(37,202)(38,203)(39,204)(40,205)(41,206)(42,207)(43,208)(44,209)(45,210)(46,211)(47,212)(48,213)(49,214)(50,215)(51,216)(52,217)(53,218)(54,219)(55,220)(56,221)(57,222)(58,223)(59,224)(60,225)(61,226)(62,227)(63,228)(64,229)(65,230)(66,231)(67,232)(68,233)(69,234)(70,235)(71,236)(72,237)(73,238)(74,239)(75,240)(76,241)(77,242)(78,243)(79,244)(80,245)(81,246)(82,247)(83,248)(84,249)(85,250)(86,251)(87,252)(88,253)(89,254)(90,255)(91,256)(92,257)(93,258)(94,259)(95,260)(96,261)(97,262)(98,263)(99,264)(100,265)(101,266)(102,267)(103,268)(104,269)(105,270)(106,271)(107,272)(108,273)(109,274)(110,275)(111,276)(112,277)(113,278)(114,279)(115,280)(116,281)(117,282)(118,283)(119,284)(120,285)(121,286)(122,287)(123,288)(124,289)(125,290)(126,291)(127,292)(128,293)(129,294)(130,295)(131,296)(132,297)(133,298)(134,299)(135,300)(136,301)(137,302)(138,303)(139,304)(140,305)(141,306)(142,307)(143,308)(144,309)(145,310)(146,311)(147,312)(148,313)(149,314)(150,315)(151,316)(152,317)(153,318)(154,319)(155,320)(156,321)(157,322)(158,323)(159,324)(160,325)(161,326)(162,327)(163,328)(164,329)(165,330)(166,331)(167,332)(168,333)(169,334)(170,335)(171,336)(172,337)(173,338)(174,339)(175,340)(176,341)(177,342)(178,343)(179,344)(180,345)(181,346)(182,347)(183,348)(184,349)(185,350)(186,351)(187,352)(188,353), (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)(189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376) );
G=PermutationGroup([[(1,354),(2,355),(3,356),(4,357),(5,358),(6,359),(7,360),(8,361),(9,362),(10,363),(11,364),(12,365),(13,366),(14,367),(15,368),(16,369),(17,370),(18,371),(19,372),(20,373),(21,374),(22,375),(23,376),(24,189),(25,190),(26,191),(27,192),(28,193),(29,194),(30,195),(31,196),(32,197),(33,198),(34,199),(35,200),(36,201),(37,202),(38,203),(39,204),(40,205),(41,206),(42,207),(43,208),(44,209),(45,210),(46,211),(47,212),(48,213),(49,214),(50,215),(51,216),(52,217),(53,218),(54,219),(55,220),(56,221),(57,222),(58,223),(59,224),(60,225),(61,226),(62,227),(63,228),(64,229),(65,230),(66,231),(67,232),(68,233),(69,234),(70,235),(71,236),(72,237),(73,238),(74,239),(75,240),(76,241),(77,242),(78,243),(79,244),(80,245),(81,246),(82,247),(83,248),(84,249),(85,250),(86,251),(87,252),(88,253),(89,254),(90,255),(91,256),(92,257),(93,258),(94,259),(95,260),(96,261),(97,262),(98,263),(99,264),(100,265),(101,266),(102,267),(103,268),(104,269),(105,270),(106,271),(107,272),(108,273),(109,274),(110,275),(111,276),(112,277),(113,278),(114,279),(115,280),(116,281),(117,282),(118,283),(119,284),(120,285),(121,286),(122,287),(123,288),(124,289),(125,290),(126,291),(127,292),(128,293),(129,294),(130,295),(131,296),(132,297),(133,298),(134,299),(135,300),(136,301),(137,302),(138,303),(139,304),(140,305),(141,306),(142,307),(143,308),(144,309),(145,310),(146,311),(147,312),(148,313),(149,314),(150,315),(151,316),(152,317),(153,318),(154,319),(155,320),(156,321),(157,322),(158,323),(159,324),(160,325),(161,326),(162,327),(163,328),(164,329),(165,330),(166,331),(167,332),(168,333),(169,334),(170,335),(171,336),(172,337),(173,338),(174,339),(175,340),(176,341),(177,342),(178,343),(179,344),(180,345),(181,346),(182,347),(183,348),(184,349),(185,350),(186,351),(187,352),(188,353)], [(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),(189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376)]])
376 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 47A | ··· | 47AT | 94A | ··· | 94EH | 188A | ··· | 188GB |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 47 | ··· | 47 | 94 | ··· | 94 | 188 | ··· | 188 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
376 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C47 | C94 | C94 | C188 |
| kernel | C2×C188 | C188 | C2×C94 | C94 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 46 | 92 | 46 | 184 |
Matrix representation of C2×C188 ►in GL2(𝔽941) generated by
| 1 | 0 |
| 0 | 940 |
| 827 | 0 |
| 0 | 827 |
G:=sub<GL(2,GF(941))| [1,0,0,940],[827,0,0,827] >;
C2×C188 in GAP, Magma, Sage, TeX
C_2\times C_{188} % in TeX
G:=Group("C2xC188"); // GroupNames label
G:=SmallGroup(376,8);
// by ID
G=gap.SmallGroup(376,8);
# by ID
G:=PCGroup([4,-2,-2,-47,-2,752]);
// Polycyclic
G:=Group<a,b|a^2=b^188=1,a*b=b*a>;
// generators/relations
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