direct product, abelian, monomial, 2-elementary
Aliases: C2×C184, SmallGroup(368,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C184 |
| C1 — C2×C184 |
| C1 — C2×C184 |
Generators and relations for C2×C184
G = < a,b | a2=b184=1, ab=ba >
Smallest permutation representation of C2×C184
►Regular action on 368 points
Generators in S368
(1 223)(2 224)(3 225)(4 226)(5 227)(6 228)(7 229)(8 230)(9 231)(10 232)(11 233)(12 234)(13 235)(14 236)(15 237)(16 238)(17 239)(18 240)(19 241)(20 242)(21 243)(22 244)(23 245)(24 246)(25 247)(26 248)(27 249)(28 250)(29 251)(30 252)(31 253)(32 254)(33 255)(34 256)(35 257)(36 258)(37 259)(38 260)(39 261)(40 262)(41 263)(42 264)(43 265)(44 266)(45 267)(46 268)(47 269)(48 270)(49 271)(50 272)(51 273)(52 274)(53 275)(54 276)(55 277)(56 278)(57 279)(58 280)(59 281)(60 282)(61 283)(62 284)(63 285)(64 286)(65 287)(66 288)(67 289)(68 290)(69 291)(70 292)(71 293)(72 294)(73 295)(74 296)(75 297)(76 298)(77 299)(78 300)(79 301)(80 302)(81 303)(82 304)(83 305)(84 306)(85 307)(86 308)(87 309)(88 310)(89 311)(90 312)(91 313)(92 314)(93 315)(94 316)(95 317)(96 318)(97 319)(98 320)(99 321)(100 322)(101 323)(102 324)(103 325)(104 326)(105 327)(106 328)(107 329)(108 330)(109 331)(110 332)(111 333)(112 334)(113 335)(114 336)(115 337)(116 338)(117 339)(118 340)(119 341)(120 342)(121 343)(122 344)(123 345)(124 346)(125 347)(126 348)(127 349)(128 350)(129 351)(130 352)(131 353)(132 354)(133 355)(134 356)(135 357)(136 358)(137 359)(138 360)(139 361)(140 362)(141 363)(142 364)(143 365)(144 366)(145 367)(146 368)(147 185)(148 186)(149 187)(150 188)(151 189)(152 190)(153 191)(154 192)(155 193)(156 194)(157 195)(158 196)(159 197)(160 198)(161 199)(162 200)(163 201)(164 202)(165 203)(166 204)(167 205)(168 206)(169 207)(170 208)(171 209)(172 210)(173 211)(174 212)(175 213)(176 214)(177 215)(178 216)(179 217)(180 218)(181 219)(182 220)(183 221)(184 222)
(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)
G:=sub<Sym(368)| (1,223)(2,224)(3,225)(4,226)(5,227)(6,228)(7,229)(8,230)(9,231)(10,232)(11,233)(12,234)(13,235)(14,236)(15,237)(16,238)(17,239)(18,240)(19,241)(20,242)(21,243)(22,244)(23,245)(24,246)(25,247)(26,248)(27,249)(28,250)(29,251)(30,252)(31,253)(32,254)(33,255)(34,256)(35,257)(36,258)(37,259)(38,260)(39,261)(40,262)(41,263)(42,264)(43,265)(44,266)(45,267)(46,268)(47,269)(48,270)(49,271)(50,272)(51,273)(52,274)(53,275)(54,276)(55,277)(56,278)(57,279)(58,280)(59,281)(60,282)(61,283)(62,284)(63,285)(64,286)(65,287)(66,288)(67,289)(68,290)(69,291)(70,292)(71,293)(72,294)(73,295)(74,296)(75,297)(76,298)(77,299)(78,300)(79,301)(80,302)(81,303)(82,304)(83,305)(84,306)(85,307)(86,308)(87,309)(88,310)(89,311)(90,312)(91,313)(92,314)(93,315)(94,316)(95,317)(96,318)(97,319)(98,320)(99,321)(100,322)(101,323)(102,324)(103,325)(104,326)(105,327)(106,328)(107,329)(108,330)(109,331)(110,332)(111,333)(112,334)(113,335)(114,336)(115,337)(116,338)(117,339)(118,340)(119,341)(120,342)(121,343)(122,344)(123,345)(124,346)(125,347)(126,348)(127,349)(128,350)(129,351)(130,352)(131,353)(132,354)(133,355)(134,356)(135,357)(136,358)(137,359)(138,360)(139,361)(140,362)(141,363)(142,364)(143,365)(144,366)(145,367)(146,368)(147,185)(148,186)(149,187)(150,188)(151,189)(152,190)(153,191)(154,192)(155,193)(156,194)(157,195)(158,196)(159,197)(160,198)(161,199)(162,200)(163,201)(164,202)(165,203)(166,204)(167,205)(168,206)(169,207)(170,208)(171,209)(172,210)(173,211)(174,212)(175,213)(176,214)(177,215)(178,216)(179,217)(180,218)(181,219)(182,220)(183,221)(184,222), (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)>;
G:=Group( (1,223)(2,224)(3,225)(4,226)(5,227)(6,228)(7,229)(8,230)(9,231)(10,232)(11,233)(12,234)(13,235)(14,236)(15,237)(16,238)(17,239)(18,240)(19,241)(20,242)(21,243)(22,244)(23,245)(24,246)(25,247)(26,248)(27,249)(28,250)(29,251)(30,252)(31,253)(32,254)(33,255)(34,256)(35,257)(36,258)(37,259)(38,260)(39,261)(40,262)(41,263)(42,264)(43,265)(44,266)(45,267)(46,268)(47,269)(48,270)(49,271)(50,272)(51,273)(52,274)(53,275)(54,276)(55,277)(56,278)(57,279)(58,280)(59,281)(60,282)(61,283)(62,284)(63,285)(64,286)(65,287)(66,288)(67,289)(68,290)(69,291)(70,292)(71,293)(72,294)(73,295)(74,296)(75,297)(76,298)(77,299)(78,300)(79,301)(80,302)(81,303)(82,304)(83,305)(84,306)(85,307)(86,308)(87,309)(88,310)(89,311)(90,312)(91,313)(92,314)(93,315)(94,316)(95,317)(96,318)(97,319)(98,320)(99,321)(100,322)(101,323)(102,324)(103,325)(104,326)(105,327)(106,328)(107,329)(108,330)(109,331)(110,332)(111,333)(112,334)(113,335)(114,336)(115,337)(116,338)(117,339)(118,340)(119,341)(120,342)(121,343)(122,344)(123,345)(124,346)(125,347)(126,348)(127,349)(128,350)(129,351)(130,352)(131,353)(132,354)(133,355)(134,356)(135,357)(136,358)(137,359)(138,360)(139,361)(140,362)(141,363)(142,364)(143,365)(144,366)(145,367)(146,368)(147,185)(148,186)(149,187)(150,188)(151,189)(152,190)(153,191)(154,192)(155,193)(156,194)(157,195)(158,196)(159,197)(160,198)(161,199)(162,200)(163,201)(164,202)(165,203)(166,204)(167,205)(168,206)(169,207)(170,208)(171,209)(172,210)(173,211)(174,212)(175,213)(176,214)(177,215)(178,216)(179,217)(180,218)(181,219)(182,220)(183,221)(184,222), (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) );
G=PermutationGroup([[(1,223),(2,224),(3,225),(4,226),(5,227),(6,228),(7,229),(8,230),(9,231),(10,232),(11,233),(12,234),(13,235),(14,236),(15,237),(16,238),(17,239),(18,240),(19,241),(20,242),(21,243),(22,244),(23,245),(24,246),(25,247),(26,248),(27,249),(28,250),(29,251),(30,252),(31,253),(32,254),(33,255),(34,256),(35,257),(36,258),(37,259),(38,260),(39,261),(40,262),(41,263),(42,264),(43,265),(44,266),(45,267),(46,268),(47,269),(48,270),(49,271),(50,272),(51,273),(52,274),(53,275),(54,276),(55,277),(56,278),(57,279),(58,280),(59,281),(60,282),(61,283),(62,284),(63,285),(64,286),(65,287),(66,288),(67,289),(68,290),(69,291),(70,292),(71,293),(72,294),(73,295),(74,296),(75,297),(76,298),(77,299),(78,300),(79,301),(80,302),(81,303),(82,304),(83,305),(84,306),(85,307),(86,308),(87,309),(88,310),(89,311),(90,312),(91,313),(92,314),(93,315),(94,316),(95,317),(96,318),(97,319),(98,320),(99,321),(100,322),(101,323),(102,324),(103,325),(104,326),(105,327),(106,328),(107,329),(108,330),(109,331),(110,332),(111,333),(112,334),(113,335),(114,336),(115,337),(116,338),(117,339),(118,340),(119,341),(120,342),(121,343),(122,344),(123,345),(124,346),(125,347),(126,348),(127,349),(128,350),(129,351),(130,352),(131,353),(132,354),(133,355),(134,356),(135,357),(136,358),(137,359),(138,360),(139,361),(140,362),(141,363),(142,364),(143,365),(144,366),(145,367),(146,368),(147,185),(148,186),(149,187),(150,188),(151,189),(152,190),(153,191),(154,192),(155,193),(156,194),(157,195),(158,196),(159,197),(160,198),(161,199),(162,200),(163,201),(164,202),(165,203),(166,204),(167,205),(168,206),(169,207),(170,208),(171,209),(172,210),(173,211),(174,212),(175,213),(176,214),(177,215),(178,216),(179,217),(180,218),(181,219),(182,220),(183,221),(184,222)], [(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)]])
368 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 23A | ··· | 23V | 46A | ··· | 46BN | 92A | ··· | 92CJ | 184A | ··· | 184FT |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 | 184 | ··· | 184 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
368 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C23 | C46 | C46 | C92 | C92 | C184 |
| kernel | C2×C184 | C184 | C2×C92 | C92 | C2×C46 | C46 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 22 | 44 | 22 | 44 | 44 | 176 |
Matrix representation of C2×C184 ►in GL2(𝔽1289) generated by
| 1288 | 0 |
| 0 | 1 |
| 373 | 0 |
| 0 | 249 |
G:=sub<GL(2,GF(1289))| [1288,0,0,1],[373,0,0,249] >;
C2×C184 in GAP, Magma, Sage, TeX
C_2\times C_{184} % in TeX
G:=Group("C2xC184"); // GroupNames label
G:=SmallGroup(368,22);
// by ID
G=gap.SmallGroup(368,22);
# by ID
G:=PCGroup([5,-2,-2,-23,-2,-2,460,58]);
// Polycyclic
G:=Group<a,b|a^2=b^184=1,a*b=b*a>;
// generators/relations
Export