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G = C32×C45order 405 = 34·5

Abelian group of type [3,3,45]

direct product, abelian, monomial, 3-elementary

Aliases: C32×C45, SmallGroup(405,11)

Series: Derived Chief Lower central Upper central

C1 — C32×C45
C1C3C15C45C3×C45 — C32×C45
C1 — C32×C45
C1 — C32×C45

Generators and relations for C32×C45
 G = < a,b,c | a3=b3=c45=1, ab=ba, ac=ca, bc=cb >

Subgroups: 100, all normal (8 characteristic)
C1, C3, C3 [×12], C5, C9 [×9], C32 [×13], C15, C15 [×12], C3×C9 [×12], C33, C45 [×9], C3×C15 [×13], C32×C9, C3×C45 [×12], C32×C15, C32×C45
Quotients: C1, C3 [×13], C5, C9 [×9], C32 [×13], C15 [×13], C3×C9 [×12], C33, C45 [×9], C3×C15 [×13], C32×C9, C3×C45 [×12], C32×C15, C32×C45

Smallest permutation representation of C32×C45
Regular action on 405 points
Generators in S405
(1 162 386)(2 163 387)(3 164 388)(4 165 389)(5 166 390)(6 167 391)(7 168 392)(8 169 393)(9 170 394)(10 171 395)(11 172 396)(12 173 397)(13 174 398)(14 175 399)(15 176 400)(16 177 401)(17 178 402)(18 179 403)(19 180 404)(20 136 405)(21 137 361)(22 138 362)(23 139 363)(24 140 364)(25 141 365)(26 142 366)(27 143 367)(28 144 368)(29 145 369)(30 146 370)(31 147 371)(32 148 372)(33 149 373)(34 150 374)(35 151 375)(36 152 376)(37 153 377)(38 154 378)(39 155 379)(40 156 380)(41 157 381)(42 158 382)(43 159 383)(44 160 384)(45 161 385)(46 123 239)(47 124 240)(48 125 241)(49 126 242)(50 127 243)(51 128 244)(52 129 245)(53 130 246)(54 131 247)(55 132 248)(56 133 249)(57 134 250)(58 135 251)(59 91 252)(60 92 253)(61 93 254)(62 94 255)(63 95 256)(64 96 257)(65 97 258)(66 98 259)(67 99 260)(68 100 261)(69 101 262)(70 102 263)(71 103 264)(72 104 265)(73 105 266)(74 106 267)(75 107 268)(76 108 269)(77 109 270)(78 110 226)(79 111 227)(80 112 228)(81 113 229)(82 114 230)(83 115 231)(84 116 232)(85 117 233)(86 118 234)(87 119 235)(88 120 236)(89 121 237)(90 122 238)(181 320 272)(182 321 273)(183 322 274)(184 323 275)(185 324 276)(186 325 277)(187 326 278)(188 327 279)(189 328 280)(190 329 281)(191 330 282)(192 331 283)(193 332 284)(194 333 285)(195 334 286)(196 335 287)(197 336 288)(198 337 289)(199 338 290)(200 339 291)(201 340 292)(202 341 293)(203 342 294)(204 343 295)(205 344 296)(206 345 297)(207 346 298)(208 347 299)(209 348 300)(210 349 301)(211 350 302)(212 351 303)(213 352 304)(214 353 305)(215 354 306)(216 355 307)(217 356 308)(218 357 309)(219 358 310)(220 359 311)(221 360 312)(222 316 313)(223 317 314)(224 318 315)(225 319 271)
(1 201 64)(2 202 65)(3 203 66)(4 204 67)(5 205 68)(6 206 69)(7 207 70)(8 208 71)(9 209 72)(10 210 73)(11 211 74)(12 212 75)(13 213 76)(14 214 77)(15 215 78)(16 216 79)(17 217 80)(18 218 81)(19 219 82)(20 220 83)(21 221 84)(22 222 85)(23 223 86)(24 224 87)(25 225 88)(26 181 89)(27 182 90)(28 183 46)(29 184 47)(30 185 48)(31 186 49)(32 187 50)(33 188 51)(34 189 52)(35 190 53)(36 191 54)(37 192 55)(38 193 56)(39 194 57)(40 195 58)(41 196 59)(42 197 60)(43 198 61)(44 199 62)(45 200 63)(91 157 335)(92 158 336)(93 159 337)(94 160 338)(95 161 339)(96 162 340)(97 163 341)(98 164 342)(99 165 343)(100 166 344)(101 167 345)(102 168 346)(103 169 347)(104 170 348)(105 171 349)(106 172 350)(107 173 351)(108 174 352)(109 175 353)(110 176 354)(111 177 355)(112 178 356)(113 179 357)(114 180 358)(115 136 359)(116 137 360)(117 138 316)(118 139 317)(119 140 318)(120 141 319)(121 142 320)(122 143 321)(123 144 322)(124 145 323)(125 146 324)(126 147 325)(127 148 326)(128 149 327)(129 150 328)(130 151 329)(131 152 330)(132 153 331)(133 154 332)(134 155 333)(135 156 334)(226 400 306)(227 401 307)(228 402 308)(229 403 309)(230 404 310)(231 405 311)(232 361 312)(233 362 313)(234 363 314)(235 364 315)(236 365 271)(237 366 272)(238 367 273)(239 368 274)(240 369 275)(241 370 276)(242 371 277)(243 372 278)(244 373 279)(245 374 280)(246 375 281)(247 376 282)(248 377 283)(249 378 284)(250 379 285)(251 380 286)(252 381 287)(253 382 288)(254 383 289)(255 384 290)(256 385 291)(257 386 292)(258 387 293)(259 388 294)(260 389 295)(261 390 296)(262 391 297)(263 392 298)(264 393 299)(265 394 300)(266 395 301)(267 396 302)(268 397 303)(269 398 304)(270 399 305)
(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)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405)

G:=sub<Sym(405)| (1,162,386)(2,163,387)(3,164,388)(4,165,389)(5,166,390)(6,167,391)(7,168,392)(8,169,393)(9,170,394)(10,171,395)(11,172,396)(12,173,397)(13,174,398)(14,175,399)(15,176,400)(16,177,401)(17,178,402)(18,179,403)(19,180,404)(20,136,405)(21,137,361)(22,138,362)(23,139,363)(24,140,364)(25,141,365)(26,142,366)(27,143,367)(28,144,368)(29,145,369)(30,146,370)(31,147,371)(32,148,372)(33,149,373)(34,150,374)(35,151,375)(36,152,376)(37,153,377)(38,154,378)(39,155,379)(40,156,380)(41,157,381)(42,158,382)(43,159,383)(44,160,384)(45,161,385)(46,123,239)(47,124,240)(48,125,241)(49,126,242)(50,127,243)(51,128,244)(52,129,245)(53,130,246)(54,131,247)(55,132,248)(56,133,249)(57,134,250)(58,135,251)(59,91,252)(60,92,253)(61,93,254)(62,94,255)(63,95,256)(64,96,257)(65,97,258)(66,98,259)(67,99,260)(68,100,261)(69,101,262)(70,102,263)(71,103,264)(72,104,265)(73,105,266)(74,106,267)(75,107,268)(76,108,269)(77,109,270)(78,110,226)(79,111,227)(80,112,228)(81,113,229)(82,114,230)(83,115,231)(84,116,232)(85,117,233)(86,118,234)(87,119,235)(88,120,236)(89,121,237)(90,122,238)(181,320,272)(182,321,273)(183,322,274)(184,323,275)(185,324,276)(186,325,277)(187,326,278)(188,327,279)(189,328,280)(190,329,281)(191,330,282)(192,331,283)(193,332,284)(194,333,285)(195,334,286)(196,335,287)(197,336,288)(198,337,289)(199,338,290)(200,339,291)(201,340,292)(202,341,293)(203,342,294)(204,343,295)(205,344,296)(206,345,297)(207,346,298)(208,347,299)(209,348,300)(210,349,301)(211,350,302)(212,351,303)(213,352,304)(214,353,305)(215,354,306)(216,355,307)(217,356,308)(218,357,309)(219,358,310)(220,359,311)(221,360,312)(222,316,313)(223,317,314)(224,318,315)(225,319,271), (1,201,64)(2,202,65)(3,203,66)(4,204,67)(5,205,68)(6,206,69)(7,207,70)(8,208,71)(9,209,72)(10,210,73)(11,211,74)(12,212,75)(13,213,76)(14,214,77)(15,215,78)(16,216,79)(17,217,80)(18,218,81)(19,219,82)(20,220,83)(21,221,84)(22,222,85)(23,223,86)(24,224,87)(25,225,88)(26,181,89)(27,182,90)(28,183,46)(29,184,47)(30,185,48)(31,186,49)(32,187,50)(33,188,51)(34,189,52)(35,190,53)(36,191,54)(37,192,55)(38,193,56)(39,194,57)(40,195,58)(41,196,59)(42,197,60)(43,198,61)(44,199,62)(45,200,63)(91,157,335)(92,158,336)(93,159,337)(94,160,338)(95,161,339)(96,162,340)(97,163,341)(98,164,342)(99,165,343)(100,166,344)(101,167,345)(102,168,346)(103,169,347)(104,170,348)(105,171,349)(106,172,350)(107,173,351)(108,174,352)(109,175,353)(110,176,354)(111,177,355)(112,178,356)(113,179,357)(114,180,358)(115,136,359)(116,137,360)(117,138,316)(118,139,317)(119,140,318)(120,141,319)(121,142,320)(122,143,321)(123,144,322)(124,145,323)(125,146,324)(126,147,325)(127,148,326)(128,149,327)(129,150,328)(130,151,329)(131,152,330)(132,153,331)(133,154,332)(134,155,333)(135,156,334)(226,400,306)(227,401,307)(228,402,308)(229,403,309)(230,404,310)(231,405,311)(232,361,312)(233,362,313)(234,363,314)(235,364,315)(236,365,271)(237,366,272)(238,367,273)(239,368,274)(240,369,275)(241,370,276)(242,371,277)(243,372,278)(244,373,279)(245,374,280)(246,375,281)(247,376,282)(248,377,283)(249,378,284)(250,379,285)(251,380,286)(252,381,287)(253,382,288)(254,383,289)(255,384,290)(256,385,291)(257,386,292)(258,387,293)(259,388,294)(260,389,295)(261,390,296)(262,391,297)(263,392,298)(264,393,299)(265,394,300)(266,395,301)(267,396,302)(268,397,303)(269,398,304)(270,399,305), (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)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405)>;

G:=Group( (1,162,386)(2,163,387)(3,164,388)(4,165,389)(5,166,390)(6,167,391)(7,168,392)(8,169,393)(9,170,394)(10,171,395)(11,172,396)(12,173,397)(13,174,398)(14,175,399)(15,176,400)(16,177,401)(17,178,402)(18,179,403)(19,180,404)(20,136,405)(21,137,361)(22,138,362)(23,139,363)(24,140,364)(25,141,365)(26,142,366)(27,143,367)(28,144,368)(29,145,369)(30,146,370)(31,147,371)(32,148,372)(33,149,373)(34,150,374)(35,151,375)(36,152,376)(37,153,377)(38,154,378)(39,155,379)(40,156,380)(41,157,381)(42,158,382)(43,159,383)(44,160,384)(45,161,385)(46,123,239)(47,124,240)(48,125,241)(49,126,242)(50,127,243)(51,128,244)(52,129,245)(53,130,246)(54,131,247)(55,132,248)(56,133,249)(57,134,250)(58,135,251)(59,91,252)(60,92,253)(61,93,254)(62,94,255)(63,95,256)(64,96,257)(65,97,258)(66,98,259)(67,99,260)(68,100,261)(69,101,262)(70,102,263)(71,103,264)(72,104,265)(73,105,266)(74,106,267)(75,107,268)(76,108,269)(77,109,270)(78,110,226)(79,111,227)(80,112,228)(81,113,229)(82,114,230)(83,115,231)(84,116,232)(85,117,233)(86,118,234)(87,119,235)(88,120,236)(89,121,237)(90,122,238)(181,320,272)(182,321,273)(183,322,274)(184,323,275)(185,324,276)(186,325,277)(187,326,278)(188,327,279)(189,328,280)(190,329,281)(191,330,282)(192,331,283)(193,332,284)(194,333,285)(195,334,286)(196,335,287)(197,336,288)(198,337,289)(199,338,290)(200,339,291)(201,340,292)(202,341,293)(203,342,294)(204,343,295)(205,344,296)(206,345,297)(207,346,298)(208,347,299)(209,348,300)(210,349,301)(211,350,302)(212,351,303)(213,352,304)(214,353,305)(215,354,306)(216,355,307)(217,356,308)(218,357,309)(219,358,310)(220,359,311)(221,360,312)(222,316,313)(223,317,314)(224,318,315)(225,319,271), (1,201,64)(2,202,65)(3,203,66)(4,204,67)(5,205,68)(6,206,69)(7,207,70)(8,208,71)(9,209,72)(10,210,73)(11,211,74)(12,212,75)(13,213,76)(14,214,77)(15,215,78)(16,216,79)(17,217,80)(18,218,81)(19,219,82)(20,220,83)(21,221,84)(22,222,85)(23,223,86)(24,224,87)(25,225,88)(26,181,89)(27,182,90)(28,183,46)(29,184,47)(30,185,48)(31,186,49)(32,187,50)(33,188,51)(34,189,52)(35,190,53)(36,191,54)(37,192,55)(38,193,56)(39,194,57)(40,195,58)(41,196,59)(42,197,60)(43,198,61)(44,199,62)(45,200,63)(91,157,335)(92,158,336)(93,159,337)(94,160,338)(95,161,339)(96,162,340)(97,163,341)(98,164,342)(99,165,343)(100,166,344)(101,167,345)(102,168,346)(103,169,347)(104,170,348)(105,171,349)(106,172,350)(107,173,351)(108,174,352)(109,175,353)(110,176,354)(111,177,355)(112,178,356)(113,179,357)(114,180,358)(115,136,359)(116,137,360)(117,138,316)(118,139,317)(119,140,318)(120,141,319)(121,142,320)(122,143,321)(123,144,322)(124,145,323)(125,146,324)(126,147,325)(127,148,326)(128,149,327)(129,150,328)(130,151,329)(131,152,330)(132,153,331)(133,154,332)(134,155,333)(135,156,334)(226,400,306)(227,401,307)(228,402,308)(229,403,309)(230,404,310)(231,405,311)(232,361,312)(233,362,313)(234,363,314)(235,364,315)(236,365,271)(237,366,272)(238,367,273)(239,368,274)(240,369,275)(241,370,276)(242,371,277)(243,372,278)(244,373,279)(245,374,280)(246,375,281)(247,376,282)(248,377,283)(249,378,284)(250,379,285)(251,380,286)(252,381,287)(253,382,288)(254,383,289)(255,384,290)(256,385,291)(257,386,292)(258,387,293)(259,388,294)(260,389,295)(261,390,296)(262,391,297)(263,392,298)(264,393,299)(265,394,300)(266,395,301)(267,396,302)(268,397,303)(269,398,304)(270,399,305), (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)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405) );

G=PermutationGroup([(1,162,386),(2,163,387),(3,164,388),(4,165,389),(5,166,390),(6,167,391),(7,168,392),(8,169,393),(9,170,394),(10,171,395),(11,172,396),(12,173,397),(13,174,398),(14,175,399),(15,176,400),(16,177,401),(17,178,402),(18,179,403),(19,180,404),(20,136,405),(21,137,361),(22,138,362),(23,139,363),(24,140,364),(25,141,365),(26,142,366),(27,143,367),(28,144,368),(29,145,369),(30,146,370),(31,147,371),(32,148,372),(33,149,373),(34,150,374),(35,151,375),(36,152,376),(37,153,377),(38,154,378),(39,155,379),(40,156,380),(41,157,381),(42,158,382),(43,159,383),(44,160,384),(45,161,385),(46,123,239),(47,124,240),(48,125,241),(49,126,242),(50,127,243),(51,128,244),(52,129,245),(53,130,246),(54,131,247),(55,132,248),(56,133,249),(57,134,250),(58,135,251),(59,91,252),(60,92,253),(61,93,254),(62,94,255),(63,95,256),(64,96,257),(65,97,258),(66,98,259),(67,99,260),(68,100,261),(69,101,262),(70,102,263),(71,103,264),(72,104,265),(73,105,266),(74,106,267),(75,107,268),(76,108,269),(77,109,270),(78,110,226),(79,111,227),(80,112,228),(81,113,229),(82,114,230),(83,115,231),(84,116,232),(85,117,233),(86,118,234),(87,119,235),(88,120,236),(89,121,237),(90,122,238),(181,320,272),(182,321,273),(183,322,274),(184,323,275),(185,324,276),(186,325,277),(187,326,278),(188,327,279),(189,328,280),(190,329,281),(191,330,282),(192,331,283),(193,332,284),(194,333,285),(195,334,286),(196,335,287),(197,336,288),(198,337,289),(199,338,290),(200,339,291),(201,340,292),(202,341,293),(203,342,294),(204,343,295),(205,344,296),(206,345,297),(207,346,298),(208,347,299),(209,348,300),(210,349,301),(211,350,302),(212,351,303),(213,352,304),(214,353,305),(215,354,306),(216,355,307),(217,356,308),(218,357,309),(219,358,310),(220,359,311),(221,360,312),(222,316,313),(223,317,314),(224,318,315),(225,319,271)], [(1,201,64),(2,202,65),(3,203,66),(4,204,67),(5,205,68),(6,206,69),(7,207,70),(8,208,71),(9,209,72),(10,210,73),(11,211,74),(12,212,75),(13,213,76),(14,214,77),(15,215,78),(16,216,79),(17,217,80),(18,218,81),(19,219,82),(20,220,83),(21,221,84),(22,222,85),(23,223,86),(24,224,87),(25,225,88),(26,181,89),(27,182,90),(28,183,46),(29,184,47),(30,185,48),(31,186,49),(32,187,50),(33,188,51),(34,189,52),(35,190,53),(36,191,54),(37,192,55),(38,193,56),(39,194,57),(40,195,58),(41,196,59),(42,197,60),(43,198,61),(44,199,62),(45,200,63),(91,157,335),(92,158,336),(93,159,337),(94,160,338),(95,161,339),(96,162,340),(97,163,341),(98,164,342),(99,165,343),(100,166,344),(101,167,345),(102,168,346),(103,169,347),(104,170,348),(105,171,349),(106,172,350),(107,173,351),(108,174,352),(109,175,353),(110,176,354),(111,177,355),(112,178,356),(113,179,357),(114,180,358),(115,136,359),(116,137,360),(117,138,316),(118,139,317),(119,140,318),(120,141,319),(121,142,320),(122,143,321),(123,144,322),(124,145,323),(125,146,324),(126,147,325),(127,148,326),(128,149,327),(129,150,328),(130,151,329),(131,152,330),(132,153,331),(133,154,332),(134,155,333),(135,156,334),(226,400,306),(227,401,307),(228,402,308),(229,403,309),(230,404,310),(231,405,311),(232,361,312),(233,362,313),(234,363,314),(235,364,315),(236,365,271),(237,366,272),(238,367,273),(239,368,274),(240,369,275),(241,370,276),(242,371,277),(243,372,278),(244,373,279),(245,374,280),(246,375,281),(247,376,282),(248,377,283),(249,378,284),(250,379,285),(251,380,286),(252,381,287),(253,382,288),(254,383,289),(255,384,290),(256,385,291),(257,386,292),(258,387,293),(259,388,294),(260,389,295),(261,390,296),(262,391,297),(263,392,298),(264,393,299),(265,394,300),(266,395,301),(267,396,302),(268,397,303),(269,398,304),(270,399,305)], [(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),(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405)])

405 conjugacy classes

class 1 3A···3Z5A5B5C5D9A···9BB15A···15CZ45A···45HH
order13···355559···915···1545···45
size11···111111···11···11···1

405 irreducible representations

dim11111111
type+
imageC1C3C3C5C9C15C15C45
kernelC32×C45C3×C45C32×C15C32×C9C3×C15C3×C9C33C32
# reps1242454968216

Matrix representation of C32×C45 in GL3(𝔽181) generated by

13200
010
001
,
13200
0480
001
,
2900
01450
0065
G:=sub<GL(3,GF(181))| [132,0,0,0,1,0,0,0,1],[132,0,0,0,48,0,0,0,1],[29,0,0,0,145,0,0,0,65] >;

C32×C45 in GAP, Magma, Sage, TeX

C_3^2\times C_{45}
% in TeX

G:=Group("C3^2xC45");
// GroupNames label

G:=SmallGroup(405,11);
// by ID

G=gap.SmallGroup(405,11);
# by ID

G:=PCGroup([5,-3,-3,-3,-5,-3,675]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^45=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽