direct product, abelian, monomial, 2-elementary
Aliases: C2×C186, SmallGroup(372,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C186 |
| C1 — C2×C186 |
| C1 — C2×C186 |
Generators and relations for C2×C186
G = < a,b | a2=b186=1, ab=ba >
Smallest permutation representation of C2×C186
►Regular action on 372 points
Generators in S372
(1 347)(2 348)(3 349)(4 350)(5 351)(6 352)(7 353)(8 354)(9 355)(10 356)(11 357)(12 358)(13 359)(14 360)(15 361)(16 362)(17 363)(18 364)(19 365)(20 366)(21 367)(22 368)(23 369)(24 370)(25 371)(26 372)(27 187)(28 188)(29 189)(30 190)(31 191)(32 192)(33 193)(34 194)(35 195)(36 196)(37 197)(38 198)(39 199)(40 200)(41 201)(42 202)(43 203)(44 204)(45 205)(46 206)(47 207)(48 208)(49 209)(50 210)(51 211)(52 212)(53 213)(54 214)(55 215)(56 216)(57 217)(58 218)(59 219)(60 220)(61 221)(62 222)(63 223)(64 224)(65 225)(66 226)(67 227)(68 228)(69 229)(70 230)(71 231)(72 232)(73 233)(74 234)(75 235)(76 236)(77 237)(78 238)(79 239)(80 240)(81 241)(82 242)(83 243)(84 244)(85 245)(86 246)(87 247)(88 248)(89 249)(90 250)(91 251)(92 252)(93 253)(94 254)(95 255)(96 256)(97 257)(98 258)(99 259)(100 260)(101 261)(102 262)(103 263)(104 264)(105 265)(106 266)(107 267)(108 268)(109 269)(110 270)(111 271)(112 272)(113 273)(114 274)(115 275)(116 276)(117 277)(118 278)(119 279)(120 280)(121 281)(122 282)(123 283)(124 284)(125 285)(126 286)(127 287)(128 288)(129 289)(130 290)(131 291)(132 292)(133 293)(134 294)(135 295)(136 296)(137 297)(138 298)(139 299)(140 300)(141 301)(142 302)(143 303)(144 304)(145 305)(146 306)(147 307)(148 308)(149 309)(150 310)(151 311)(152 312)(153 313)(154 314)(155 315)(156 316)(157 317)(158 318)(159 319)(160 320)(161 321)(162 322)(163 323)(164 324)(165 325)(166 326)(167 327)(168 328)(169 329)(170 330)(171 331)(172 332)(173 333)(174 334)(175 335)(176 336)(177 337)(178 338)(179 339)(180 340)(181 341)(182 342)(183 343)(184 344)(185 345)(186 346)
(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)
G:=sub<Sym(372)| (1,347)(2,348)(3,349)(4,350)(5,351)(6,352)(7,353)(8,354)(9,355)(10,356)(11,357)(12,358)(13,359)(14,360)(15,361)(16,362)(17,363)(18,364)(19,365)(20,366)(21,367)(22,368)(23,369)(24,370)(25,371)(26,372)(27,187)(28,188)(29,189)(30,190)(31,191)(32,192)(33,193)(34,194)(35,195)(36,196)(37,197)(38,198)(39,199)(40,200)(41,201)(42,202)(43,203)(44,204)(45,205)(46,206)(47,207)(48,208)(49,209)(50,210)(51,211)(52,212)(53,213)(54,214)(55,215)(56,216)(57,217)(58,218)(59,219)(60,220)(61,221)(62,222)(63,223)(64,224)(65,225)(66,226)(67,227)(68,228)(69,229)(70,230)(71,231)(72,232)(73,233)(74,234)(75,235)(76,236)(77,237)(78,238)(79,239)(80,240)(81,241)(82,242)(83,243)(84,244)(85,245)(86,246)(87,247)(88,248)(89,249)(90,250)(91,251)(92,252)(93,253)(94,254)(95,255)(96,256)(97,257)(98,258)(99,259)(100,260)(101,261)(102,262)(103,263)(104,264)(105,265)(106,266)(107,267)(108,268)(109,269)(110,270)(111,271)(112,272)(113,273)(114,274)(115,275)(116,276)(117,277)(118,278)(119,279)(120,280)(121,281)(122,282)(123,283)(124,284)(125,285)(126,286)(127,287)(128,288)(129,289)(130,290)(131,291)(132,292)(133,293)(134,294)(135,295)(136,296)(137,297)(138,298)(139,299)(140,300)(141,301)(142,302)(143,303)(144,304)(145,305)(146,306)(147,307)(148,308)(149,309)(150,310)(151,311)(152,312)(153,313)(154,314)(155,315)(156,316)(157,317)(158,318)(159,319)(160,320)(161,321)(162,322)(163,323)(164,324)(165,325)(166,326)(167,327)(168,328)(169,329)(170,330)(171,331)(172,332)(173,333)(174,334)(175,335)(176,336)(177,337)(178,338)(179,339)(180,340)(181,341)(182,342)(183,343)(184,344)(185,345)(186,346), (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)>;
G:=Group( (1,347)(2,348)(3,349)(4,350)(5,351)(6,352)(7,353)(8,354)(9,355)(10,356)(11,357)(12,358)(13,359)(14,360)(15,361)(16,362)(17,363)(18,364)(19,365)(20,366)(21,367)(22,368)(23,369)(24,370)(25,371)(26,372)(27,187)(28,188)(29,189)(30,190)(31,191)(32,192)(33,193)(34,194)(35,195)(36,196)(37,197)(38,198)(39,199)(40,200)(41,201)(42,202)(43,203)(44,204)(45,205)(46,206)(47,207)(48,208)(49,209)(50,210)(51,211)(52,212)(53,213)(54,214)(55,215)(56,216)(57,217)(58,218)(59,219)(60,220)(61,221)(62,222)(63,223)(64,224)(65,225)(66,226)(67,227)(68,228)(69,229)(70,230)(71,231)(72,232)(73,233)(74,234)(75,235)(76,236)(77,237)(78,238)(79,239)(80,240)(81,241)(82,242)(83,243)(84,244)(85,245)(86,246)(87,247)(88,248)(89,249)(90,250)(91,251)(92,252)(93,253)(94,254)(95,255)(96,256)(97,257)(98,258)(99,259)(100,260)(101,261)(102,262)(103,263)(104,264)(105,265)(106,266)(107,267)(108,268)(109,269)(110,270)(111,271)(112,272)(113,273)(114,274)(115,275)(116,276)(117,277)(118,278)(119,279)(120,280)(121,281)(122,282)(123,283)(124,284)(125,285)(126,286)(127,287)(128,288)(129,289)(130,290)(131,291)(132,292)(133,293)(134,294)(135,295)(136,296)(137,297)(138,298)(139,299)(140,300)(141,301)(142,302)(143,303)(144,304)(145,305)(146,306)(147,307)(148,308)(149,309)(150,310)(151,311)(152,312)(153,313)(154,314)(155,315)(156,316)(157,317)(158,318)(159,319)(160,320)(161,321)(162,322)(163,323)(164,324)(165,325)(166,326)(167,327)(168,328)(169,329)(170,330)(171,331)(172,332)(173,333)(174,334)(175,335)(176,336)(177,337)(178,338)(179,339)(180,340)(181,341)(182,342)(183,343)(184,344)(185,345)(186,346), (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) );
G=PermutationGroup([[(1,347),(2,348),(3,349),(4,350),(5,351),(6,352),(7,353),(8,354),(9,355),(10,356),(11,357),(12,358),(13,359),(14,360),(15,361),(16,362),(17,363),(18,364),(19,365),(20,366),(21,367),(22,368),(23,369),(24,370),(25,371),(26,372),(27,187),(28,188),(29,189),(30,190),(31,191),(32,192),(33,193),(34,194),(35,195),(36,196),(37,197),(38,198),(39,199),(40,200),(41,201),(42,202),(43,203),(44,204),(45,205),(46,206),(47,207),(48,208),(49,209),(50,210),(51,211),(52,212),(53,213),(54,214),(55,215),(56,216),(57,217),(58,218),(59,219),(60,220),(61,221),(62,222),(63,223),(64,224),(65,225),(66,226),(67,227),(68,228),(69,229),(70,230),(71,231),(72,232),(73,233),(74,234),(75,235),(76,236),(77,237),(78,238),(79,239),(80,240),(81,241),(82,242),(83,243),(84,244),(85,245),(86,246),(87,247),(88,248),(89,249),(90,250),(91,251),(92,252),(93,253),(94,254),(95,255),(96,256),(97,257),(98,258),(99,259),(100,260),(101,261),(102,262),(103,263),(104,264),(105,265),(106,266),(107,267),(108,268),(109,269),(110,270),(111,271),(112,272),(113,273),(114,274),(115,275),(116,276),(117,277),(118,278),(119,279),(120,280),(121,281),(122,282),(123,283),(124,284),(125,285),(126,286),(127,287),(128,288),(129,289),(130,290),(131,291),(132,292),(133,293),(134,294),(135,295),(136,296),(137,297),(138,298),(139,299),(140,300),(141,301),(142,302),(143,303),(144,304),(145,305),(146,306),(147,307),(148,308),(149,309),(150,310),(151,311),(152,312),(153,313),(154,314),(155,315),(156,316),(157,317),(158,318),(159,319),(160,320),(161,321),(162,322),(163,323),(164,324),(165,325),(166,326),(167,327),(168,328),(169,329),(170,330),(171,331),(172,332),(173,333),(174,334),(175,335),(176,336),(177,337),(178,338),(179,339),(180,340),(181,341),(182,342),(183,343),(184,344),(185,345),(186,346)], [(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)]])
372 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 31A | ··· | 31AD | 62A | ··· | 62CL | 93A | ··· | 93BH | 186A | ··· | 186FX |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 31 | ··· | 31 | 62 | ··· | 62 | 93 | ··· | 93 | 186 | ··· | 186 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
372 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C31 | C62 | C93 | C186 |
| kernel | C2×C186 | C186 | C2×C62 | C62 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 30 | 90 | 60 | 180 |
Matrix representation of C2×C186 ►in GL2(𝔽373) generated by
| 372 | 0 |
| 0 | 372 |
| 372 | 0 |
| 0 | 9 |
G:=sub<GL(2,GF(373))| [372,0,0,372],[372,0,0,9] >;
C2×C186 in GAP, Magma, Sage, TeX
C_2\times C_{186} % in TeX
G:=Group("C2xC186"); // GroupNames label
G:=SmallGroup(372,15);
// by ID
G=gap.SmallGroup(372,15);
# by ID
G:=PCGroup([4,-2,-2,-3,-31]);
// Polycyclic
G:=Group<a,b|a^2=b^186=1,a*b=b*a>;
// generators/relations
Export