direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic45, C90⋊3C4, C18⋊Dic5, C2.2D90, C22.D45, C10⋊2Dic9, C30.43D6, C6.11D30, C18.11D10, C10.11D18, C6.2Dic15, C30.6Dic3, C90.11C22, (C2×C18).D5, (C2×C10).D9, C45⋊10(C2×C4), C5⋊3(C2×Dic9), C9⋊2(C2×Dic5), (C2×C30).3S3, (C2×C90).1C2, (C2×C6).2D15, C3.(C2×Dic15), C15.3(C2×Dic3), SmallGroup(360,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C45 — C2×Dic45 |
Generators and relations for C2×Dic45
G = < a,b,c | a2=b90=1, c2=b45, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic45
►Regular action on 360 points
Generators in S360
(1 105)(2 106)(3 107)(4 108)(5 109)(6 110)(7 111)(8 112)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 141)(38 142)(39 143)(40 144)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 161)(58 162)(59 163)(60 164)(61 165)(62 166)(63 167)(64 168)(65 169)(66 170)(67 171)(68 172)(69 173)(70 174)(71 175)(72 176)(73 177)(74 178)(75 179)(76 180)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(181 300)(182 301)(183 302)(184 303)(185 304)(186 305)(187 306)(188 307)(189 308)(190 309)(191 310)(192 311)(193 312)(194 313)(195 314)(196 315)(197 316)(198 317)(199 318)(200 319)(201 320)(202 321)(203 322)(204 323)(205 324)(206 325)(207 326)(208 327)(209 328)(210 329)(211 330)(212 331)(213 332)(214 333)(215 334)(216 335)(217 336)(218 337)(219 338)(220 339)(221 340)(222 341)(223 342)(224 343)(225 344)(226 345)(227 346)(228 347)(229 348)(230 349)(231 350)(232 351)(233 352)(234 353)(235 354)(236 355)(237 356)(238 357)(239 358)(240 359)(241 360)(242 271)(243 272)(244 273)(245 274)(246 275)(247 276)(248 277)(249 278)(250 279)(251 280)(252 281)(253 282)(254 283)(255 284)(256 285)(257 286)(258 287)(259 288)(260 289)(261 290)(262 291)(263 292)(264 293)(265 294)(266 295)(267 296)(268 297)(269 298)(270 299)
(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)
(1 324 46 279)(2 323 47 278)(3 322 48 277)(4 321 49 276)(5 320 50 275)(6 319 51 274)(7 318 52 273)(8 317 53 272)(9 316 54 271)(10 315 55 360)(11 314 56 359)(12 313 57 358)(13 312 58 357)(14 311 59 356)(15 310 60 355)(16 309 61 354)(17 308 62 353)(18 307 63 352)(19 306 64 351)(20 305 65 350)(21 304 66 349)(22 303 67 348)(23 302 68 347)(24 301 69 346)(25 300 70 345)(26 299 71 344)(27 298 72 343)(28 297 73 342)(29 296 74 341)(30 295 75 340)(31 294 76 339)(32 293 77 338)(33 292 78 337)(34 291 79 336)(35 290 80 335)(36 289 81 334)(37 288 82 333)(38 287 83 332)(39 286 84 331)(40 285 85 330)(41 284 86 329)(42 283 87 328)(43 282 88 327)(44 281 89 326)(45 280 90 325)(91 219 136 264)(92 218 137 263)(93 217 138 262)(94 216 139 261)(95 215 140 260)(96 214 141 259)(97 213 142 258)(98 212 143 257)(99 211 144 256)(100 210 145 255)(101 209 146 254)(102 208 147 253)(103 207 148 252)(104 206 149 251)(105 205 150 250)(106 204 151 249)(107 203 152 248)(108 202 153 247)(109 201 154 246)(110 200 155 245)(111 199 156 244)(112 198 157 243)(113 197 158 242)(114 196 159 241)(115 195 160 240)(116 194 161 239)(117 193 162 238)(118 192 163 237)(119 191 164 236)(120 190 165 235)(121 189 166 234)(122 188 167 233)(123 187 168 232)(124 186 169 231)(125 185 170 230)(126 184 171 229)(127 183 172 228)(128 182 173 227)(129 181 174 226)(130 270 175 225)(131 269 176 224)(132 268 177 223)(133 267 178 222)(134 266 179 221)(135 265 180 220)
G:=sub<Sym(360)| (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,141)(38,142)(39,143)(40,144)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,161)(58,162)(59,163)(60,164)(61,165)(62,166)(63,167)(64,168)(65,169)(66,170)(67,171)(68,172)(69,173)(70,174)(71,175)(72,176)(73,177)(74,178)(75,179)(76,180)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(181,300)(182,301)(183,302)(184,303)(185,304)(186,305)(187,306)(188,307)(189,308)(190,309)(191,310)(192,311)(193,312)(194,313)(195,314)(196,315)(197,316)(198,317)(199,318)(200,319)(201,320)(202,321)(203,322)(204,323)(205,324)(206,325)(207,326)(208,327)(209,328)(210,329)(211,330)(212,331)(213,332)(214,333)(215,334)(216,335)(217,336)(218,337)(219,338)(220,339)(221,340)(222,341)(223,342)(224,343)(225,344)(226,345)(227,346)(228,347)(229,348)(230,349)(231,350)(232,351)(233,352)(234,353)(235,354)(236,355)(237,356)(238,357)(239,358)(240,359)(241,360)(242,271)(243,272)(244,273)(245,274)(246,275)(247,276)(248,277)(249,278)(250,279)(251,280)(252,281)(253,282)(254,283)(255,284)(256,285)(257,286)(258,287)(259,288)(260,289)(261,290)(262,291)(263,292)(264,293)(265,294)(266,295)(267,296)(268,297)(269,298)(270,299), (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), (1,324,46,279)(2,323,47,278)(3,322,48,277)(4,321,49,276)(5,320,50,275)(6,319,51,274)(7,318,52,273)(8,317,53,272)(9,316,54,271)(10,315,55,360)(11,314,56,359)(12,313,57,358)(13,312,58,357)(14,311,59,356)(15,310,60,355)(16,309,61,354)(17,308,62,353)(18,307,63,352)(19,306,64,351)(20,305,65,350)(21,304,66,349)(22,303,67,348)(23,302,68,347)(24,301,69,346)(25,300,70,345)(26,299,71,344)(27,298,72,343)(28,297,73,342)(29,296,74,341)(30,295,75,340)(31,294,76,339)(32,293,77,338)(33,292,78,337)(34,291,79,336)(35,290,80,335)(36,289,81,334)(37,288,82,333)(38,287,83,332)(39,286,84,331)(40,285,85,330)(41,284,86,329)(42,283,87,328)(43,282,88,327)(44,281,89,326)(45,280,90,325)(91,219,136,264)(92,218,137,263)(93,217,138,262)(94,216,139,261)(95,215,140,260)(96,214,141,259)(97,213,142,258)(98,212,143,257)(99,211,144,256)(100,210,145,255)(101,209,146,254)(102,208,147,253)(103,207,148,252)(104,206,149,251)(105,205,150,250)(106,204,151,249)(107,203,152,248)(108,202,153,247)(109,201,154,246)(110,200,155,245)(111,199,156,244)(112,198,157,243)(113,197,158,242)(114,196,159,241)(115,195,160,240)(116,194,161,239)(117,193,162,238)(118,192,163,237)(119,191,164,236)(120,190,165,235)(121,189,166,234)(122,188,167,233)(123,187,168,232)(124,186,169,231)(125,185,170,230)(126,184,171,229)(127,183,172,228)(128,182,173,227)(129,181,174,226)(130,270,175,225)(131,269,176,224)(132,268,177,223)(133,267,178,222)(134,266,179,221)(135,265,180,220)>;
G:=Group( (1,105)(2,106)(3,107)(4,108)(5,109)(6,110)(7,111)(8,112)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,141)(38,142)(39,143)(40,144)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,161)(58,162)(59,163)(60,164)(61,165)(62,166)(63,167)(64,168)(65,169)(66,170)(67,171)(68,172)(69,173)(70,174)(71,175)(72,176)(73,177)(74,178)(75,179)(76,180)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(181,300)(182,301)(183,302)(184,303)(185,304)(186,305)(187,306)(188,307)(189,308)(190,309)(191,310)(192,311)(193,312)(194,313)(195,314)(196,315)(197,316)(198,317)(199,318)(200,319)(201,320)(202,321)(203,322)(204,323)(205,324)(206,325)(207,326)(208,327)(209,328)(210,329)(211,330)(212,331)(213,332)(214,333)(215,334)(216,335)(217,336)(218,337)(219,338)(220,339)(221,340)(222,341)(223,342)(224,343)(225,344)(226,345)(227,346)(228,347)(229,348)(230,349)(231,350)(232,351)(233,352)(234,353)(235,354)(236,355)(237,356)(238,357)(239,358)(240,359)(241,360)(242,271)(243,272)(244,273)(245,274)(246,275)(247,276)(248,277)(249,278)(250,279)(251,280)(252,281)(253,282)(254,283)(255,284)(256,285)(257,286)(258,287)(259,288)(260,289)(261,290)(262,291)(263,292)(264,293)(265,294)(266,295)(267,296)(268,297)(269,298)(270,299), (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), (1,324,46,279)(2,323,47,278)(3,322,48,277)(4,321,49,276)(5,320,50,275)(6,319,51,274)(7,318,52,273)(8,317,53,272)(9,316,54,271)(10,315,55,360)(11,314,56,359)(12,313,57,358)(13,312,58,357)(14,311,59,356)(15,310,60,355)(16,309,61,354)(17,308,62,353)(18,307,63,352)(19,306,64,351)(20,305,65,350)(21,304,66,349)(22,303,67,348)(23,302,68,347)(24,301,69,346)(25,300,70,345)(26,299,71,344)(27,298,72,343)(28,297,73,342)(29,296,74,341)(30,295,75,340)(31,294,76,339)(32,293,77,338)(33,292,78,337)(34,291,79,336)(35,290,80,335)(36,289,81,334)(37,288,82,333)(38,287,83,332)(39,286,84,331)(40,285,85,330)(41,284,86,329)(42,283,87,328)(43,282,88,327)(44,281,89,326)(45,280,90,325)(91,219,136,264)(92,218,137,263)(93,217,138,262)(94,216,139,261)(95,215,140,260)(96,214,141,259)(97,213,142,258)(98,212,143,257)(99,211,144,256)(100,210,145,255)(101,209,146,254)(102,208,147,253)(103,207,148,252)(104,206,149,251)(105,205,150,250)(106,204,151,249)(107,203,152,248)(108,202,153,247)(109,201,154,246)(110,200,155,245)(111,199,156,244)(112,198,157,243)(113,197,158,242)(114,196,159,241)(115,195,160,240)(116,194,161,239)(117,193,162,238)(118,192,163,237)(119,191,164,236)(120,190,165,235)(121,189,166,234)(122,188,167,233)(123,187,168,232)(124,186,169,231)(125,185,170,230)(126,184,171,229)(127,183,172,228)(128,182,173,227)(129,181,174,226)(130,270,175,225)(131,269,176,224)(132,268,177,223)(133,267,178,222)(134,266,179,221)(135,265,180,220) );
G=PermutationGroup([[(1,105),(2,106),(3,107),(4,108),(5,109),(6,110),(7,111),(8,112),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,121),(18,122),(19,123),(20,124),(21,125),(22,126),(23,127),(24,128),(25,129),(26,130),(27,131),(28,132),(29,133),(30,134),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,141),(38,142),(39,143),(40,144),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,154),(51,155),(52,156),(53,157),(54,158),(55,159),(56,160),(57,161),(58,162),(59,163),(60,164),(61,165),(62,166),(63,167),(64,168),(65,169),(66,170),(67,171),(68,172),(69,173),(70,174),(71,175),(72,176),(73,177),(74,178),(75,179),(76,180),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(181,300),(182,301),(183,302),(184,303),(185,304),(186,305),(187,306),(188,307),(189,308),(190,309),(191,310),(192,311),(193,312),(194,313),(195,314),(196,315),(197,316),(198,317),(199,318),(200,319),(201,320),(202,321),(203,322),(204,323),(205,324),(206,325),(207,326),(208,327),(209,328),(210,329),(211,330),(212,331),(213,332),(214,333),(215,334),(216,335),(217,336),(218,337),(219,338),(220,339),(221,340),(222,341),(223,342),(224,343),(225,344),(226,345),(227,346),(228,347),(229,348),(230,349),(231,350),(232,351),(233,352),(234,353),(235,354),(236,355),(237,356),(238,357),(239,358),(240,359),(241,360),(242,271),(243,272),(244,273),(245,274),(246,275),(247,276),(248,277),(249,278),(250,279),(251,280),(252,281),(253,282),(254,283),(255,284),(256,285),(257,286),(258,287),(259,288),(260,289),(261,290),(262,291),(263,292),(264,293),(265,294),(266,295),(267,296),(268,297),(269,298),(270,299)], [(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)], [(1,324,46,279),(2,323,47,278),(3,322,48,277),(4,321,49,276),(5,320,50,275),(6,319,51,274),(7,318,52,273),(8,317,53,272),(9,316,54,271),(10,315,55,360),(11,314,56,359),(12,313,57,358),(13,312,58,357),(14,311,59,356),(15,310,60,355),(16,309,61,354),(17,308,62,353),(18,307,63,352),(19,306,64,351),(20,305,65,350),(21,304,66,349),(22,303,67,348),(23,302,68,347),(24,301,69,346),(25,300,70,345),(26,299,71,344),(27,298,72,343),(28,297,73,342),(29,296,74,341),(30,295,75,340),(31,294,76,339),(32,293,77,338),(33,292,78,337),(34,291,79,336),(35,290,80,335),(36,289,81,334),(37,288,82,333),(38,287,83,332),(39,286,84,331),(40,285,85,330),(41,284,86,329),(42,283,87,328),(43,282,88,327),(44,281,89,326),(45,280,90,325),(91,219,136,264),(92,218,137,263),(93,217,138,262),(94,216,139,261),(95,215,140,260),(96,214,141,259),(97,213,142,258),(98,212,143,257),(99,211,144,256),(100,210,145,255),(101,209,146,254),(102,208,147,253),(103,207,148,252),(104,206,149,251),(105,205,150,250),(106,204,151,249),(107,203,152,248),(108,202,153,247),(109,201,154,246),(110,200,155,245),(111,199,156,244),(112,198,157,243),(113,197,158,242),(114,196,159,241),(115,195,160,240),(116,194,161,239),(117,193,162,238),(118,192,163,237),(119,191,164,236),(120,190,165,235),(121,189,166,234),(122,188,167,233),(123,187,168,232),(124,186,169,231),(125,185,170,230),(126,184,171,229),(127,183,172,228),(128,182,173,227),(129,181,174,226),(130,270,175,225),(131,269,176,224),(132,268,177,223),(133,267,178,222),(134,266,179,221),(135,265,180,220)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 9A | 9B | 9C | 10A | ··· | 10F | 15A | 15B | 15C | 15D | 18A | ··· | 18I | 30A | ··· | 30L | 45A | ··· | 45L | 90A | ··· | 90AJ |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 9 | 9 | 9 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 18 | ··· | 18 | 30 | ··· | 30 | 45 | ··· | 45 | 90 | ··· | 90 |
| size | 1 | 1 | 1 | 1 | 2 | 45 | 45 | 45 | 45 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | + | - | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | D5 | Dic3 | D6 | D9 | Dic5 | D10 | D15 | Dic9 | D18 | Dic15 | D30 | D45 | Dic45 | D90 |
| kernel | C2×Dic45 | Dic45 | C2×C90 | C90 | C2×C30 | C2×C18 | C30 | C30 | C2×C10 | C18 | C18 | C2×C6 | C10 | C10 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 2 | 1 | 3 | 4 | 2 | 4 | 6 | 3 | 8 | 4 | 12 | 24 | 12 |
Matrix representation of C2×Dic45 ►in GL3(𝔽181) generated by
| 180 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 93 | 129 |
| 0 | 52 | 41 |
| 180 | 0 | 0 |
| 0 | 0 | 19 |
| 0 | 19 | 0 |
G:=sub<GL(3,GF(181))| [180,0,0,0,1,0,0,0,1],[1,0,0,0,93,52,0,129,41],[180,0,0,0,0,19,0,19,0] >;
C2×Dic45 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{45} % in TeX
G:=Group("C2xDic45"); // GroupNames label
G:=SmallGroup(360,28);
// by ID
G=gap.SmallGroup(360,28);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-3,24,3267,741,2884,8645]);
// Polycyclic
G:=Group<a,b,c|a^2=b^90=1,c^2=b^45,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export