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G = C22×C90order 360 = 23·32·5

Abelian group of type [2,2,90]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C90, SmallGroup(360,50)

Series: Derived Chief Lower central Upper central

C1 — C22×C90
C1C3C15C45C90C2×C90 — C22×C90
C1 — C22×C90
C1 — C22×C90

Generators and relations for C22×C90
 G = < a,b,c | a2=b2=c90=1, ab=ba, ac=ca, bc=cb >

Subgroups: 96, all normal (12 characteristic)
C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], C23, C9, C10 [×7], C2×C6 [×7], C15, C18 [×7], C2×C10 [×7], C22×C6, C30 [×7], C2×C18 [×7], C22×C10, C45, C2×C30 [×7], C22×C18, C90 [×7], C22×C30, C2×C90 [×7], C22×C90
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], C23, C9, C10 [×7], C2×C6 [×7], C15, C18 [×7], C2×C10 [×7], C22×C6, C30 [×7], C2×C18 [×7], C22×C10, C45, C2×C30 [×7], C22×C18, C90 [×7], C22×C30, C2×C90 [×7], C22×C90

Smallest permutation representation of C22×C90
Regular action on 360 points
Generators in S360
(1 241)(2 242)(3 243)(4 244)(5 245)(6 246)(7 247)(8 248)(9 249)(10 250)(11 251)(12 252)(13 253)(14 254)(15 255)(16 256)(17 257)(18 258)(19 259)(20 260)(21 261)(22 262)(23 263)(24 264)(25 265)(26 266)(27 267)(28 268)(29 269)(30 270)(31 181)(32 182)(33 183)(34 184)(35 185)(36 186)(37 187)(38 188)(39 189)(40 190)(41 191)(42 192)(43 193)(44 194)(45 195)(46 196)(47 197)(48 198)(49 199)(50 200)(51 201)(52 202)(53 203)(54 204)(55 205)(56 206)(57 207)(58 208)(59 209)(60 210)(61 211)(62 212)(63 213)(64 214)(65 215)(66 216)(67 217)(68 218)(69 219)(70 220)(71 221)(72 222)(73 223)(74 224)(75 225)(76 226)(77 227)(78 228)(79 229)(80 230)(81 231)(82 232)(83 233)(84 234)(85 235)(86 236)(87 237)(88 238)(89 239)(90 240)(91 312)(92 313)(93 314)(94 315)(95 316)(96 317)(97 318)(98 319)(99 320)(100 321)(101 322)(102 323)(103 324)(104 325)(105 326)(106 327)(107 328)(108 329)(109 330)(110 331)(111 332)(112 333)(113 334)(114 335)(115 336)(116 337)(117 338)(118 339)(119 340)(120 341)(121 342)(122 343)(123 344)(124 345)(125 346)(126 347)(127 348)(128 349)(129 350)(130 351)(131 352)(132 353)(133 354)(134 355)(135 356)(136 357)(137 358)(138 359)(139 360)(140 271)(141 272)(142 273)(143 274)(144 275)(145 276)(146 277)(147 278)(148 279)(149 280)(150 281)(151 282)(152 283)(153 284)(154 285)(155 286)(156 287)(157 288)(158 289)(159 290)(160 291)(161 292)(162 293)(163 294)(164 295)(165 296)(166 297)(167 298)(168 299)(169 300)(170 301)(171 302)(172 303)(173 304)(174 305)(175 306)(176 307)(177 308)(178 309)(179 310)(180 311)
(1 166)(2 167)(3 168)(4 169)(5 170)(6 171)(7 172)(8 173)(9 174)(10 175)(11 176)(12 177)(13 178)(14 179)(15 180)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 140)(66 141)(67 142)(68 143)(69 144)(70 145)(71 146)(72 147)(73 148)(74 149)(75 150)(76 151)(77 152)(78 153)(79 154)(80 155)(81 156)(82 157)(83 158)(84 159)(85 160)(86 161)(87 162)(88 163)(89 164)(90 165)(181 327)(182 328)(183 329)(184 330)(185 331)(186 332)(187 333)(188 334)(189 335)(190 336)(191 337)(192 338)(193 339)(194 340)(195 341)(196 342)(197 343)(198 344)(199 345)(200 346)(201 347)(202 348)(203 349)(204 350)(205 351)(206 352)(207 353)(208 354)(209 355)(210 356)(211 357)(212 358)(213 359)(214 360)(215 271)(216 272)(217 273)(218 274)(219 275)(220 276)(221 277)(222 278)(223 279)(224 280)(225 281)(226 282)(227 283)(228 284)(229 285)(230 286)(231 287)(232 288)(233 289)(234 290)(235 291)(236 292)(237 293)(238 294)(239 295)(240 296)(241 297)(242 298)(243 299)(244 300)(245 301)(246 302)(247 303)(248 304)(249 305)(250 306)(251 307)(252 308)(253 309)(254 310)(255 311)(256 312)(257 313)(258 314)(259 315)(260 316)(261 317)(262 318)(263 319)(264 320)(265 321)(266 322)(267 323)(268 324)(269 325)(270 326)
(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)

G:=sub<Sym(360)| (1,241)(2,242)(3,243)(4,244)(5,245)(6,246)(7,247)(8,248)(9,249)(10,250)(11,251)(12,252)(13,253)(14,254)(15,255)(16,256)(17,257)(18,258)(19,259)(20,260)(21,261)(22,262)(23,263)(24,264)(25,265)(26,266)(27,267)(28,268)(29,269)(30,270)(31,181)(32,182)(33,183)(34,184)(35,185)(36,186)(37,187)(38,188)(39,189)(40,190)(41,191)(42,192)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,199)(50,200)(51,201)(52,202)(53,203)(54,204)(55,205)(56,206)(57,207)(58,208)(59,209)(60,210)(61,211)(62,212)(63,213)(64,214)(65,215)(66,216)(67,217)(68,218)(69,219)(70,220)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,227)(78,228)(79,229)(80,230)(81,231)(82,232)(83,233)(84,234)(85,235)(86,236)(87,237)(88,238)(89,239)(90,240)(91,312)(92,313)(93,314)(94,315)(95,316)(96,317)(97,318)(98,319)(99,320)(100,321)(101,322)(102,323)(103,324)(104,325)(105,326)(106,327)(107,328)(108,329)(109,330)(110,331)(111,332)(112,333)(113,334)(114,335)(115,336)(116,337)(117,338)(118,339)(119,340)(120,341)(121,342)(122,343)(123,344)(124,345)(125,346)(126,347)(127,348)(128,349)(129,350)(130,351)(131,352)(132,353)(133,354)(134,355)(135,356)(136,357)(137,358)(138,359)(139,360)(140,271)(141,272)(142,273)(143,274)(144,275)(145,276)(146,277)(147,278)(148,279)(149,280)(150,281)(151,282)(152,283)(153,284)(154,285)(155,286)(156,287)(157,288)(158,289)(159,290)(160,291)(161,292)(162,293)(163,294)(164,295)(165,296)(166,297)(167,298)(168,299)(169,300)(170,301)(171,302)(172,303)(173,304)(174,305)(175,306)(176,307)(177,308)(178,309)(179,310)(180,311), (1,166)(2,167)(3,168)(4,169)(5,170)(6,171)(7,172)(8,173)(9,174)(10,175)(11,176)(12,177)(13,178)(14,179)(15,180)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(181,327)(182,328)(183,329)(184,330)(185,331)(186,332)(187,333)(188,334)(189,335)(190,336)(191,337)(192,338)(193,339)(194,340)(195,341)(196,342)(197,343)(198,344)(199,345)(200,346)(201,347)(202,348)(203,349)(204,350)(205,351)(206,352)(207,353)(208,354)(209,355)(210,356)(211,357)(212,358)(213,359)(214,360)(215,271)(216,272)(217,273)(218,274)(219,275)(220,276)(221,277)(222,278)(223,279)(224,280)(225,281)(226,282)(227,283)(228,284)(229,285)(230,286)(231,287)(232,288)(233,289)(234,290)(235,291)(236,292)(237,293)(238,294)(239,295)(240,296)(241,297)(242,298)(243,299)(244,300)(245,301)(246,302)(247,303)(248,304)(249,305)(250,306)(251,307)(252,308)(253,309)(254,310)(255,311)(256,312)(257,313)(258,314)(259,315)(260,316)(261,317)(262,318)(263,319)(264,320)(265,321)(266,322)(267,323)(268,324)(269,325)(270,326), (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)>;

G:=Group( (1,241)(2,242)(3,243)(4,244)(5,245)(6,246)(7,247)(8,248)(9,249)(10,250)(11,251)(12,252)(13,253)(14,254)(15,255)(16,256)(17,257)(18,258)(19,259)(20,260)(21,261)(22,262)(23,263)(24,264)(25,265)(26,266)(27,267)(28,268)(29,269)(30,270)(31,181)(32,182)(33,183)(34,184)(35,185)(36,186)(37,187)(38,188)(39,189)(40,190)(41,191)(42,192)(43,193)(44,194)(45,195)(46,196)(47,197)(48,198)(49,199)(50,200)(51,201)(52,202)(53,203)(54,204)(55,205)(56,206)(57,207)(58,208)(59,209)(60,210)(61,211)(62,212)(63,213)(64,214)(65,215)(66,216)(67,217)(68,218)(69,219)(70,220)(71,221)(72,222)(73,223)(74,224)(75,225)(76,226)(77,227)(78,228)(79,229)(80,230)(81,231)(82,232)(83,233)(84,234)(85,235)(86,236)(87,237)(88,238)(89,239)(90,240)(91,312)(92,313)(93,314)(94,315)(95,316)(96,317)(97,318)(98,319)(99,320)(100,321)(101,322)(102,323)(103,324)(104,325)(105,326)(106,327)(107,328)(108,329)(109,330)(110,331)(111,332)(112,333)(113,334)(114,335)(115,336)(116,337)(117,338)(118,339)(119,340)(120,341)(121,342)(122,343)(123,344)(124,345)(125,346)(126,347)(127,348)(128,349)(129,350)(130,351)(131,352)(132,353)(133,354)(134,355)(135,356)(136,357)(137,358)(138,359)(139,360)(140,271)(141,272)(142,273)(143,274)(144,275)(145,276)(146,277)(147,278)(148,279)(149,280)(150,281)(151,282)(152,283)(153,284)(154,285)(155,286)(156,287)(157,288)(158,289)(159,290)(160,291)(161,292)(162,293)(163,294)(164,295)(165,296)(166,297)(167,298)(168,299)(169,300)(170,301)(171,302)(172,303)(173,304)(174,305)(175,306)(176,307)(177,308)(178,309)(179,310)(180,311), (1,166)(2,167)(3,168)(4,169)(5,170)(6,171)(7,172)(8,173)(9,174)(10,175)(11,176)(12,177)(13,178)(14,179)(15,180)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,145)(71,146)(72,147)(73,148)(74,149)(75,150)(76,151)(77,152)(78,153)(79,154)(80,155)(81,156)(82,157)(83,158)(84,159)(85,160)(86,161)(87,162)(88,163)(89,164)(90,165)(181,327)(182,328)(183,329)(184,330)(185,331)(186,332)(187,333)(188,334)(189,335)(190,336)(191,337)(192,338)(193,339)(194,340)(195,341)(196,342)(197,343)(198,344)(199,345)(200,346)(201,347)(202,348)(203,349)(204,350)(205,351)(206,352)(207,353)(208,354)(209,355)(210,356)(211,357)(212,358)(213,359)(214,360)(215,271)(216,272)(217,273)(218,274)(219,275)(220,276)(221,277)(222,278)(223,279)(224,280)(225,281)(226,282)(227,283)(228,284)(229,285)(230,286)(231,287)(232,288)(233,289)(234,290)(235,291)(236,292)(237,293)(238,294)(239,295)(240,296)(241,297)(242,298)(243,299)(244,300)(245,301)(246,302)(247,303)(248,304)(249,305)(250,306)(251,307)(252,308)(253,309)(254,310)(255,311)(256,312)(257,313)(258,314)(259,315)(260,316)(261,317)(262,318)(263,319)(264,320)(265,321)(266,322)(267,323)(268,324)(269,325)(270,326), (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) );

G=PermutationGroup([(1,241),(2,242),(3,243),(4,244),(5,245),(6,246),(7,247),(8,248),(9,249),(10,250),(11,251),(12,252),(13,253),(14,254),(15,255),(16,256),(17,257),(18,258),(19,259),(20,260),(21,261),(22,262),(23,263),(24,264),(25,265),(26,266),(27,267),(28,268),(29,269),(30,270),(31,181),(32,182),(33,183),(34,184),(35,185),(36,186),(37,187),(38,188),(39,189),(40,190),(41,191),(42,192),(43,193),(44,194),(45,195),(46,196),(47,197),(48,198),(49,199),(50,200),(51,201),(52,202),(53,203),(54,204),(55,205),(56,206),(57,207),(58,208),(59,209),(60,210),(61,211),(62,212),(63,213),(64,214),(65,215),(66,216),(67,217),(68,218),(69,219),(70,220),(71,221),(72,222),(73,223),(74,224),(75,225),(76,226),(77,227),(78,228),(79,229),(80,230),(81,231),(82,232),(83,233),(84,234),(85,235),(86,236),(87,237),(88,238),(89,239),(90,240),(91,312),(92,313),(93,314),(94,315),(95,316),(96,317),(97,318),(98,319),(99,320),(100,321),(101,322),(102,323),(103,324),(104,325),(105,326),(106,327),(107,328),(108,329),(109,330),(110,331),(111,332),(112,333),(113,334),(114,335),(115,336),(116,337),(117,338),(118,339),(119,340),(120,341),(121,342),(122,343),(123,344),(124,345),(125,346),(126,347),(127,348),(128,349),(129,350),(130,351),(131,352),(132,353),(133,354),(134,355),(135,356),(136,357),(137,358),(138,359),(139,360),(140,271),(141,272),(142,273),(143,274),(144,275),(145,276),(146,277),(147,278),(148,279),(149,280),(150,281),(151,282),(152,283),(153,284),(154,285),(155,286),(156,287),(157,288),(158,289),(159,290),(160,291),(161,292),(162,293),(163,294),(164,295),(165,296),(166,297),(167,298),(168,299),(169,300),(170,301),(171,302),(172,303),(173,304),(174,305),(175,306),(176,307),(177,308),(178,309),(179,310),(180,311)], [(1,166),(2,167),(3,168),(4,169),(5,170),(6,171),(7,172),(8,173),(9,174),(10,175),(11,176),(12,177),(13,178),(14,179),(15,180),(16,91),(17,92),(18,93),(19,94),(20,95),(21,96),(22,97),(23,98),(24,99),(25,100),(26,101),(27,102),(28,103),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,113),(39,114),(40,115),(41,116),(42,117),(43,118),(44,119),(45,120),(46,121),(47,122),(48,123),(49,124),(50,125),(51,126),(52,127),(53,128),(54,129),(55,130),(56,131),(57,132),(58,133),(59,134),(60,135),(61,136),(62,137),(63,138),(64,139),(65,140),(66,141),(67,142),(68,143),(69,144),(70,145),(71,146),(72,147),(73,148),(74,149),(75,150),(76,151),(77,152),(78,153),(79,154),(80,155),(81,156),(82,157),(83,158),(84,159),(85,160),(86,161),(87,162),(88,163),(89,164),(90,165),(181,327),(182,328),(183,329),(184,330),(185,331),(186,332),(187,333),(188,334),(189,335),(190,336),(191,337),(192,338),(193,339),(194,340),(195,341),(196,342),(197,343),(198,344),(199,345),(200,346),(201,347),(202,348),(203,349),(204,350),(205,351),(206,352),(207,353),(208,354),(209,355),(210,356),(211,357),(212,358),(213,359),(214,360),(215,271),(216,272),(217,273),(218,274),(219,275),(220,276),(221,277),(222,278),(223,279),(224,280),(225,281),(226,282),(227,283),(228,284),(229,285),(230,286),(231,287),(232,288),(233,289),(234,290),(235,291),(236,292),(237,293),(238,294),(239,295),(240,296),(241,297),(242,298),(243,299),(244,300),(245,301),(246,302),(247,303),(248,304),(249,305),(250,306),(251,307),(252,308),(253,309),(254,310),(255,311),(256,312),(257,313),(258,314),(259,315),(260,316),(261,317),(262,318),(263,319),(264,320),(265,321),(266,322),(267,323),(268,324),(269,325),(270,326)], [(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)])

360 conjugacy classes

class 1 2A···2G3A3B5A5B5C5D6A···6N9A···9F10A···10AB15A···15H18A···18AP30A···30BD45A···45X90A···90FL
order12···23355556···69···910···1015···1518···1830···3045···4590···90
size11···11111111···11···11···11···11···11···11···11···1

360 irreducible representations

dim111111111111
type++
imageC1C2C3C5C6C9C10C15C18C30C45C90
kernelC22×C90C2×C90C22×C30C22×C18C2×C30C22×C10C2×C18C22×C6C2×C10C2×C6C23C22
# reps1724146288425624168

Matrix representation of C22×C90 in GL3(𝔽181) generated by

100
01800
001
,
100
010
00180
,
11600
0750
0013
G:=sub<GL(3,GF(181))| [1,0,0,0,180,0,0,0,1],[1,0,0,0,1,0,0,0,180],[116,0,0,0,75,0,0,0,13] >;

C22×C90 in GAP, Magma, Sage, TeX

C_2^2\times C_{90}
% in TeX

G:=Group("C2^2xC90");
// GroupNames label

G:=SmallGroup(360,50);
// by ID

G=gap.SmallGroup(360,50);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,381]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^90=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽