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## G = C3×C5⋊C32order 480 = 25·3·5

### Direct product of C3 and C5⋊C32

Aliases: C3×C5⋊C32, C5⋊C96, C152C32, C10.C48, C60.5C8, C24.9F5, C20.2C24, C40.3C12, C30.2C16, C120.8C4, C8.4(C3×F5), C6.2(C5⋊C16), C12.5(C5⋊C8), C52C16.2C6, C2.(C3×C5⋊C16), C4.2(C3×C5⋊C8), (C3×C52C16).4C2, SmallGroup(480,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C3×C5⋊C32
 Chief series C1 — C5 — C10 — C20 — C40 — C5⋊2C16 — C3×C5⋊2C16 — C3×C5⋊C32
 Lower central C5 — C3×C5⋊C32
 Upper central C1 — C24

Generators and relations for C3×C5⋊C32
G = < a,b,c | a3=b5=c32=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C3×C5⋊C32
Regular action on 480 points
Generators in S480
(1 392 467)(2 393 468)(3 394 469)(4 395 470)(5 396 471)(6 397 472)(7 398 473)(8 399 474)(9 400 475)(10 401 476)(11 402 477)(12 403 478)(13 404 479)(14 405 480)(15 406 449)(16 407 450)(17 408 451)(18 409 452)(19 410 453)(20 411 454)(21 412 455)(22 413 456)(23 414 457)(24 415 458)(25 416 459)(26 385 460)(27 386 461)(28 387 462)(29 388 463)(30 389 464)(31 390 465)(32 391 466)(33 152 262)(34 153 263)(35 154 264)(36 155 265)(37 156 266)(38 157 267)(39 158 268)(40 159 269)(41 160 270)(42 129 271)(43 130 272)(44 131 273)(45 132 274)(46 133 275)(47 134 276)(48 135 277)(49 136 278)(50 137 279)(51 138 280)(52 139 281)(53 140 282)(54 141 283)(55 142 284)(56 143 285)(57 144 286)(58 145 287)(59 146 288)(60 147 257)(61 148 258)(62 149 259)(63 150 260)(64 151 261)(65 291 442)(66 292 443)(67 293 444)(68 294 445)(69 295 446)(70 296 447)(71 297 448)(72 298 417)(73 299 418)(74 300 419)(75 301 420)(76 302 421)(77 303 422)(78 304 423)(79 305 424)(80 306 425)(81 307 426)(82 308 427)(83 309 428)(84 310 429)(85 311 430)(86 312 431)(87 313 432)(88 314 433)(89 315 434)(90 316 435)(91 317 436)(92 318 437)(93 319 438)(94 320 439)(95 289 440)(96 290 441)(97 256 357)(98 225 358)(99 226 359)(100 227 360)(101 228 361)(102 229 362)(103 230 363)(104 231 364)(105 232 365)(106 233 366)(107 234 367)(108 235 368)(109 236 369)(110 237 370)(111 238 371)(112 239 372)(113 240 373)(114 241 374)(115 242 375)(116 243 376)(117 244 377)(118 245 378)(119 246 379)(120 247 380)(121 248 381)(122 249 382)(123 250 383)(124 251 384)(125 252 353)(126 253 354)(127 254 355)(128 255 356)(161 348 213)(162 349 214)(163 350 215)(164 351 216)(165 352 217)(166 321 218)(167 322 219)(168 323 220)(169 324 221)(170 325 222)(171 326 223)(172 327 224)(173 328 193)(174 329 194)(175 330 195)(176 331 196)(177 332 197)(178 333 198)(179 334 199)(180 335 200)(181 336 201)(182 337 202)(183 338 203)(184 339 204)(185 340 205)(186 341 206)(187 342 207)(188 343 208)(189 344 209)(190 345 210)(191 346 211)(192 347 212)
(1 306 265 165 380)(2 166 307 381 266)(3 382 167 267 308)(4 268 383 309 168)(5 310 269 169 384)(6 170 311 353 270)(7 354 171 271 312)(8 272 355 313 172)(9 314 273 173 356)(10 174 315 357 274)(11 358 175 275 316)(12 276 359 317 176)(13 318 277 177 360)(14 178 319 361 278)(15 362 179 279 320)(16 280 363 289 180)(17 290 281 181 364)(18 182 291 365 282)(19 366 183 283 292)(20 284 367 293 184)(21 294 285 185 368)(22 186 295 369 286)(23 370 187 287 296)(24 288 371 297 188)(25 298 257 189 372)(26 190 299 373 258)(27 374 191 259 300)(28 260 375 301 192)(29 302 261 161 376)(30 162 303 377 262)(31 378 163 263 304)(32 264 379 305 164)(33 389 349 422 117)(34 423 390 118 350)(35 119 424 351 391)(36 352 120 392 425)(37 393 321 426 121)(38 427 394 122 322)(39 123 428 323 395)(40 324 124 396 429)(41 397 325 430 125)(42 431 398 126 326)(43 127 432 327 399)(44 328 128 400 433)(45 401 329 434 97)(46 435 402 98 330)(47 99 436 331 403)(48 332 100 404 437)(49 405 333 438 101)(50 439 406 102 334)(51 103 440 335 407)(52 336 104 408 441)(53 409 337 442 105)(54 443 410 106 338)(55 107 444 339 411)(56 340 108 412 445)(57 413 341 446 109)(58 447 414 110 342)(59 111 448 343 415)(60 344 112 416 417)(61 385 345 418 113)(62 419 386 114 346)(63 115 420 347 387)(64 348 116 388 421)(65 232 140 452 202)(66 453 233 203 141)(67 204 454 142 234)(68 143 205 235 455)(69 236 144 456 206)(70 457 237 207 145)(71 208 458 146 238)(72 147 209 239 459)(73 240 148 460 210)(74 461 241 211 149)(75 212 462 150 242)(76 151 213 243 463)(77 244 152 464 214)(78 465 245 215 153)(79 216 466 154 246)(80 155 217 247 467)(81 248 156 468 218)(82 469 249 219 157)(83 220 470 158 250)(84 159 221 251 471)(85 252 160 472 222)(86 473 253 223 129)(87 224 474 130 254)(88 131 193 255 475)(89 256 132 476 194)(90 477 225 195 133)(91 196 478 134 226)(92 135 197 227 479)(93 228 136 480 198)(94 449 229 199 137)(95 200 450 138 230)(96 139 201 231 451)
(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)(385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416)(417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448)(449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480)

G:=sub<Sym(480)| (1,392,467)(2,393,468)(3,394,469)(4,395,470)(5,396,471)(6,397,472)(7,398,473)(8,399,474)(9,400,475)(10,401,476)(11,402,477)(12,403,478)(13,404,479)(14,405,480)(15,406,449)(16,407,450)(17,408,451)(18,409,452)(19,410,453)(20,411,454)(21,412,455)(22,413,456)(23,414,457)(24,415,458)(25,416,459)(26,385,460)(27,386,461)(28,387,462)(29,388,463)(30,389,464)(31,390,465)(32,391,466)(33,152,262)(34,153,263)(35,154,264)(36,155,265)(37,156,266)(38,157,267)(39,158,268)(40,159,269)(41,160,270)(42,129,271)(43,130,272)(44,131,273)(45,132,274)(46,133,275)(47,134,276)(48,135,277)(49,136,278)(50,137,279)(51,138,280)(52,139,281)(53,140,282)(54,141,283)(55,142,284)(56,143,285)(57,144,286)(58,145,287)(59,146,288)(60,147,257)(61,148,258)(62,149,259)(63,150,260)(64,151,261)(65,291,442)(66,292,443)(67,293,444)(68,294,445)(69,295,446)(70,296,447)(71,297,448)(72,298,417)(73,299,418)(74,300,419)(75,301,420)(76,302,421)(77,303,422)(78,304,423)(79,305,424)(80,306,425)(81,307,426)(82,308,427)(83,309,428)(84,310,429)(85,311,430)(86,312,431)(87,313,432)(88,314,433)(89,315,434)(90,316,435)(91,317,436)(92,318,437)(93,319,438)(94,320,439)(95,289,440)(96,290,441)(97,256,357)(98,225,358)(99,226,359)(100,227,360)(101,228,361)(102,229,362)(103,230,363)(104,231,364)(105,232,365)(106,233,366)(107,234,367)(108,235,368)(109,236,369)(110,237,370)(111,238,371)(112,239,372)(113,240,373)(114,241,374)(115,242,375)(116,243,376)(117,244,377)(118,245,378)(119,246,379)(120,247,380)(121,248,381)(122,249,382)(123,250,383)(124,251,384)(125,252,353)(126,253,354)(127,254,355)(128,255,356)(161,348,213)(162,349,214)(163,350,215)(164,351,216)(165,352,217)(166,321,218)(167,322,219)(168,323,220)(169,324,221)(170,325,222)(171,326,223)(172,327,224)(173,328,193)(174,329,194)(175,330,195)(176,331,196)(177,332,197)(178,333,198)(179,334,199)(180,335,200)(181,336,201)(182,337,202)(183,338,203)(184,339,204)(185,340,205)(186,341,206)(187,342,207)(188,343,208)(189,344,209)(190,345,210)(191,346,211)(192,347,212), (1,306,265,165,380)(2,166,307,381,266)(3,382,167,267,308)(4,268,383,309,168)(5,310,269,169,384)(6,170,311,353,270)(7,354,171,271,312)(8,272,355,313,172)(9,314,273,173,356)(10,174,315,357,274)(11,358,175,275,316)(12,276,359,317,176)(13,318,277,177,360)(14,178,319,361,278)(15,362,179,279,320)(16,280,363,289,180)(17,290,281,181,364)(18,182,291,365,282)(19,366,183,283,292)(20,284,367,293,184)(21,294,285,185,368)(22,186,295,369,286)(23,370,187,287,296)(24,288,371,297,188)(25,298,257,189,372)(26,190,299,373,258)(27,374,191,259,300)(28,260,375,301,192)(29,302,261,161,376)(30,162,303,377,262)(31,378,163,263,304)(32,264,379,305,164)(33,389,349,422,117)(34,423,390,118,350)(35,119,424,351,391)(36,352,120,392,425)(37,393,321,426,121)(38,427,394,122,322)(39,123,428,323,395)(40,324,124,396,429)(41,397,325,430,125)(42,431,398,126,326)(43,127,432,327,399)(44,328,128,400,433)(45,401,329,434,97)(46,435,402,98,330)(47,99,436,331,403)(48,332,100,404,437)(49,405,333,438,101)(50,439,406,102,334)(51,103,440,335,407)(52,336,104,408,441)(53,409,337,442,105)(54,443,410,106,338)(55,107,444,339,411)(56,340,108,412,445)(57,413,341,446,109)(58,447,414,110,342)(59,111,448,343,415)(60,344,112,416,417)(61,385,345,418,113)(62,419,386,114,346)(63,115,420,347,387)(64,348,116,388,421)(65,232,140,452,202)(66,453,233,203,141)(67,204,454,142,234)(68,143,205,235,455)(69,236,144,456,206)(70,457,237,207,145)(71,208,458,146,238)(72,147,209,239,459)(73,240,148,460,210)(74,461,241,211,149)(75,212,462,150,242)(76,151,213,243,463)(77,244,152,464,214)(78,465,245,215,153)(79,216,466,154,246)(80,155,217,247,467)(81,248,156,468,218)(82,469,249,219,157)(83,220,470,158,250)(84,159,221,251,471)(85,252,160,472,222)(86,473,253,223,129)(87,224,474,130,254)(88,131,193,255,475)(89,256,132,476,194)(90,477,225,195,133)(91,196,478,134,226)(92,135,197,227,479)(93,228,136,480,198)(94,449,229,199,137)(95,200,450,138,230)(96,139,201,231,451), (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)(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416)(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)(449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480)>;

G:=Group( (1,392,467)(2,393,468)(3,394,469)(4,395,470)(5,396,471)(6,397,472)(7,398,473)(8,399,474)(9,400,475)(10,401,476)(11,402,477)(12,403,478)(13,404,479)(14,405,480)(15,406,449)(16,407,450)(17,408,451)(18,409,452)(19,410,453)(20,411,454)(21,412,455)(22,413,456)(23,414,457)(24,415,458)(25,416,459)(26,385,460)(27,386,461)(28,387,462)(29,388,463)(30,389,464)(31,390,465)(32,391,466)(33,152,262)(34,153,263)(35,154,264)(36,155,265)(37,156,266)(38,157,267)(39,158,268)(40,159,269)(41,160,270)(42,129,271)(43,130,272)(44,131,273)(45,132,274)(46,133,275)(47,134,276)(48,135,277)(49,136,278)(50,137,279)(51,138,280)(52,139,281)(53,140,282)(54,141,283)(55,142,284)(56,143,285)(57,144,286)(58,145,287)(59,146,288)(60,147,257)(61,148,258)(62,149,259)(63,150,260)(64,151,261)(65,291,442)(66,292,443)(67,293,444)(68,294,445)(69,295,446)(70,296,447)(71,297,448)(72,298,417)(73,299,418)(74,300,419)(75,301,420)(76,302,421)(77,303,422)(78,304,423)(79,305,424)(80,306,425)(81,307,426)(82,308,427)(83,309,428)(84,310,429)(85,311,430)(86,312,431)(87,313,432)(88,314,433)(89,315,434)(90,316,435)(91,317,436)(92,318,437)(93,319,438)(94,320,439)(95,289,440)(96,290,441)(97,256,357)(98,225,358)(99,226,359)(100,227,360)(101,228,361)(102,229,362)(103,230,363)(104,231,364)(105,232,365)(106,233,366)(107,234,367)(108,235,368)(109,236,369)(110,237,370)(111,238,371)(112,239,372)(113,240,373)(114,241,374)(115,242,375)(116,243,376)(117,244,377)(118,245,378)(119,246,379)(120,247,380)(121,248,381)(122,249,382)(123,250,383)(124,251,384)(125,252,353)(126,253,354)(127,254,355)(128,255,356)(161,348,213)(162,349,214)(163,350,215)(164,351,216)(165,352,217)(166,321,218)(167,322,219)(168,323,220)(169,324,221)(170,325,222)(171,326,223)(172,327,224)(173,328,193)(174,329,194)(175,330,195)(176,331,196)(177,332,197)(178,333,198)(179,334,199)(180,335,200)(181,336,201)(182,337,202)(183,338,203)(184,339,204)(185,340,205)(186,341,206)(187,342,207)(188,343,208)(189,344,209)(190,345,210)(191,346,211)(192,347,212), (1,306,265,165,380)(2,166,307,381,266)(3,382,167,267,308)(4,268,383,309,168)(5,310,269,169,384)(6,170,311,353,270)(7,354,171,271,312)(8,272,355,313,172)(9,314,273,173,356)(10,174,315,357,274)(11,358,175,275,316)(12,276,359,317,176)(13,318,277,177,360)(14,178,319,361,278)(15,362,179,279,320)(16,280,363,289,180)(17,290,281,181,364)(18,182,291,365,282)(19,366,183,283,292)(20,284,367,293,184)(21,294,285,185,368)(22,186,295,369,286)(23,370,187,287,296)(24,288,371,297,188)(25,298,257,189,372)(26,190,299,373,258)(27,374,191,259,300)(28,260,375,301,192)(29,302,261,161,376)(30,162,303,377,262)(31,378,163,263,304)(32,264,379,305,164)(33,389,349,422,117)(34,423,390,118,350)(35,119,424,351,391)(36,352,120,392,425)(37,393,321,426,121)(38,427,394,122,322)(39,123,428,323,395)(40,324,124,396,429)(41,397,325,430,125)(42,431,398,126,326)(43,127,432,327,399)(44,328,128,400,433)(45,401,329,434,97)(46,435,402,98,330)(47,99,436,331,403)(48,332,100,404,437)(49,405,333,438,101)(50,439,406,102,334)(51,103,440,335,407)(52,336,104,408,441)(53,409,337,442,105)(54,443,410,106,338)(55,107,444,339,411)(56,340,108,412,445)(57,413,341,446,109)(58,447,414,110,342)(59,111,448,343,415)(60,344,112,416,417)(61,385,345,418,113)(62,419,386,114,346)(63,115,420,347,387)(64,348,116,388,421)(65,232,140,452,202)(66,453,233,203,141)(67,204,454,142,234)(68,143,205,235,455)(69,236,144,456,206)(70,457,237,207,145)(71,208,458,146,238)(72,147,209,239,459)(73,240,148,460,210)(74,461,241,211,149)(75,212,462,150,242)(76,151,213,243,463)(77,244,152,464,214)(78,465,245,215,153)(79,216,466,154,246)(80,155,217,247,467)(81,248,156,468,218)(82,469,249,219,157)(83,220,470,158,250)(84,159,221,251,471)(85,252,160,472,222)(86,473,253,223,129)(87,224,474,130,254)(88,131,193,255,475)(89,256,132,476,194)(90,477,225,195,133)(91,196,478,134,226)(92,135,197,227,479)(93,228,136,480,198)(94,449,229,199,137)(95,200,450,138,230)(96,139,201,231,451), (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)(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416)(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)(449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480) );

G=PermutationGroup([[(1,392,467),(2,393,468),(3,394,469),(4,395,470),(5,396,471),(6,397,472),(7,398,473),(8,399,474),(9,400,475),(10,401,476),(11,402,477),(12,403,478),(13,404,479),(14,405,480),(15,406,449),(16,407,450),(17,408,451),(18,409,452),(19,410,453),(20,411,454),(21,412,455),(22,413,456),(23,414,457),(24,415,458),(25,416,459),(26,385,460),(27,386,461),(28,387,462),(29,388,463),(30,389,464),(31,390,465),(32,391,466),(33,152,262),(34,153,263),(35,154,264),(36,155,265),(37,156,266),(38,157,267),(39,158,268),(40,159,269),(41,160,270),(42,129,271),(43,130,272),(44,131,273),(45,132,274),(46,133,275),(47,134,276),(48,135,277),(49,136,278),(50,137,279),(51,138,280),(52,139,281),(53,140,282),(54,141,283),(55,142,284),(56,143,285),(57,144,286),(58,145,287),(59,146,288),(60,147,257),(61,148,258),(62,149,259),(63,150,260),(64,151,261),(65,291,442),(66,292,443),(67,293,444),(68,294,445),(69,295,446),(70,296,447),(71,297,448),(72,298,417),(73,299,418),(74,300,419),(75,301,420),(76,302,421),(77,303,422),(78,304,423),(79,305,424),(80,306,425),(81,307,426),(82,308,427),(83,309,428),(84,310,429),(85,311,430),(86,312,431),(87,313,432),(88,314,433),(89,315,434),(90,316,435),(91,317,436),(92,318,437),(93,319,438),(94,320,439),(95,289,440),(96,290,441),(97,256,357),(98,225,358),(99,226,359),(100,227,360),(101,228,361),(102,229,362),(103,230,363),(104,231,364),(105,232,365),(106,233,366),(107,234,367),(108,235,368),(109,236,369),(110,237,370),(111,238,371),(112,239,372),(113,240,373),(114,241,374),(115,242,375),(116,243,376),(117,244,377),(118,245,378),(119,246,379),(120,247,380),(121,248,381),(122,249,382),(123,250,383),(124,251,384),(125,252,353),(126,253,354),(127,254,355),(128,255,356),(161,348,213),(162,349,214),(163,350,215),(164,351,216),(165,352,217),(166,321,218),(167,322,219),(168,323,220),(169,324,221),(170,325,222),(171,326,223),(172,327,224),(173,328,193),(174,329,194),(175,330,195),(176,331,196),(177,332,197),(178,333,198),(179,334,199),(180,335,200),(181,336,201),(182,337,202),(183,338,203),(184,339,204),(185,340,205),(186,341,206),(187,342,207),(188,343,208),(189,344,209),(190,345,210),(191,346,211),(192,347,212)], [(1,306,265,165,380),(2,166,307,381,266),(3,382,167,267,308),(4,268,383,309,168),(5,310,269,169,384),(6,170,311,353,270),(7,354,171,271,312),(8,272,355,313,172),(9,314,273,173,356),(10,174,315,357,274),(11,358,175,275,316),(12,276,359,317,176),(13,318,277,177,360),(14,178,319,361,278),(15,362,179,279,320),(16,280,363,289,180),(17,290,281,181,364),(18,182,291,365,282),(19,366,183,283,292),(20,284,367,293,184),(21,294,285,185,368),(22,186,295,369,286),(23,370,187,287,296),(24,288,371,297,188),(25,298,257,189,372),(26,190,299,373,258),(27,374,191,259,300),(28,260,375,301,192),(29,302,261,161,376),(30,162,303,377,262),(31,378,163,263,304),(32,264,379,305,164),(33,389,349,422,117),(34,423,390,118,350),(35,119,424,351,391),(36,352,120,392,425),(37,393,321,426,121),(38,427,394,122,322),(39,123,428,323,395),(40,324,124,396,429),(41,397,325,430,125),(42,431,398,126,326),(43,127,432,327,399),(44,328,128,400,433),(45,401,329,434,97),(46,435,402,98,330),(47,99,436,331,403),(48,332,100,404,437),(49,405,333,438,101),(50,439,406,102,334),(51,103,440,335,407),(52,336,104,408,441),(53,409,337,442,105),(54,443,410,106,338),(55,107,444,339,411),(56,340,108,412,445),(57,413,341,446,109),(58,447,414,110,342),(59,111,448,343,415),(60,344,112,416,417),(61,385,345,418,113),(62,419,386,114,346),(63,115,420,347,387),(64,348,116,388,421),(65,232,140,452,202),(66,453,233,203,141),(67,204,454,142,234),(68,143,205,235,455),(69,236,144,456,206),(70,457,237,207,145),(71,208,458,146,238),(72,147,209,239,459),(73,240,148,460,210),(74,461,241,211,149),(75,212,462,150,242),(76,151,213,243,463),(77,244,152,464,214),(78,465,245,215,153),(79,216,466,154,246),(80,155,217,247,467),(81,248,156,468,218),(82,469,249,219,157),(83,220,470,158,250),(84,159,221,251,471),(85,252,160,472,222),(86,473,253,223,129),(87,224,474,130,254),(88,131,193,255,475),(89,256,132,476,194),(90,477,225,195,133),(91,196,478,134,226),(92,135,197,227,479),(93,228,136,480,198),(94,449,229,199,137),(95,200,450,138,230),(96,139,201,231,451)], [(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),(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416),(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448),(449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480)]])

120 conjugacy classes

 class 1 2 3A 3B 4A 4B 5 6A 6B 8A 8B 8C 8D 10 12A 12B 12C 12D 15A 15B 16A ··· 16H 20A 20B 24A ··· 24H 30A 30B 32A ··· 32P 40A 40B 40C 40D 48A ··· 48P 60A 60B 60C 60D 96A ··· 96AF 120A ··· 120H order 1 2 3 3 4 4 5 6 6 8 8 8 8 10 12 12 12 12 15 15 16 ··· 16 20 20 24 ··· 24 30 30 32 ··· 32 40 40 40 40 48 ··· 48 60 60 60 60 96 ··· 96 120 ··· 120 size 1 1 1 1 1 1 4 1 1 1 1 1 1 4 1 1 1 1 4 4 5 ··· 5 4 4 1 ··· 1 4 4 5 ··· 5 4 4 4 4 5 ··· 5 4 4 4 4 5 ··· 5 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 type + + + - image C1 C2 C3 C4 C6 C8 C12 C16 C24 C32 C48 C96 F5 C5⋊C8 C3×F5 C5⋊C16 C3×C5⋊C8 C5⋊C32 C3×C5⋊C16 C3×C5⋊C32 kernel C3×C5⋊C32 C3×C5⋊2C16 C5⋊C32 C120 C5⋊2C16 C60 C40 C30 C20 C15 C10 C5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 8 16 16 32 1 1 2 2 2 4 4 8

Matrix representation of C3×C5⋊C32 in GL5(𝔽3361)

 1 0 0 0 0 0 2468 0 0 0 0 0 2468 0 0 0 0 0 2468 0 0 0 0 0 2468
,
 1 0 0 0 0 0 0 0 0 3360 0 1 0 0 3360 0 0 1 0 3360 0 0 0 1 3360
,
 1066 0 0 0 0 0 2286 284 1284 1687 0 209 1971 2465 612 0 1390 896 2749 1896 0 1674 2180 1075 3077

G:=sub<GL(5,GF(3361))| [1,0,0,0,0,0,2468,0,0,0,0,0,2468,0,0,0,0,0,2468,0,0,0,0,0,2468],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,3360,3360,3360,3360],[1066,0,0,0,0,0,2286,209,1390,1674,0,284,1971,896,2180,0,1284,2465,2749,1075,0,1687,612,1896,3077] >;

C3×C5⋊C32 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes C_{32}
% in TeX

G:=Group("C3xC5:C32");
// GroupNames label

G:=SmallGroup(480,5);
// by ID

G=gap.SmallGroup(480,5);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,-2,-5,42,58,80,102,9414,3156]);
// Polycyclic

G:=Group<a,b,c|a^3=b^5=c^32=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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