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## G = C22×C90order 360 = 23·32·5

### Abelian group of type [2,2,90]

Aliases: C22×C90, SmallGroup(360,50)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C90
 Chief series C1 — C3 — C15 — C45 — C90 — C2×C90 — C22×C90
 Lower central C1 — C22×C90
 Upper central 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, C3, C22, C5, C6, C23, C9, C10, C2×C6, C15, C18, C2×C10, C22×C6, C30, C2×C18, C22×C10, C45, C2×C30, C22×C18, C90, C22×C30, C2×C90, C22×C90
Quotients: C1, C2, C3, C22, C5, C6, C23, C9, C10, C2×C6, C15, C18, C2×C10, C22×C6, C30, C2×C18, C22×C10, C45, C2×C30, C22×C18, C90, C22×C30, C2×C90, C22×C90

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

360 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C5 C6 C9 C10 C15 C18 C30 C45 C90 kernel C22×C90 C2×C90 C22×C30 C22×C18 C2×C30 C22×C10 C2×C18 C22×C6 C2×C10 C2×C6 C23 C22 # reps 1 7 2 4 14 6 28 8 42 56 24 168

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

 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
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