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## G = C32×C45order 405 = 34·5

### Abelian group of type [3,3,45]

Aliases: C32×C45, SmallGroup(405,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C45
 Chief series C1 — C3 — C15 — C45 — C3×C45 — C32×C45
 Lower central C1 — C32×C45
 Upper central 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, C5, C9, C32, C15, C15, C3×C9, C33, C45, C3×C15, C32×C9, C3×C45, C32×C15, C32×C45
Quotients: C1, C3, C5, C9, C32, C15, C3×C9, C33, C45, C3×C15, C32×C9, C3×C45, C32×C15, C32×C45

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

405 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + image C1 C3 C3 C5 C9 C15 C15 C45 kernel C32×C45 C3×C45 C32×C15 C32×C9 C3×C15 C3×C9 C33 C32 # reps 1 24 2 4 54 96 8 216

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

 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
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

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