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## G = C3×C5⋊2C32order 480 = 25·3·5

### Direct product of C3 and C5⋊2C32

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C3×C5⋊2C32
 Chief series C1 — C5 — C10 — C20 — C40 — C80 — C240 — C3×C5⋊2C32
 Lower central C5 — C3×C5⋊2C32
 Upper central C1 — C48

Generators and relations for C3×C52C32
G = < a,b,c | a3=b5=c32=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×C52C32
Regular action on 480 points
Generators in S480
(1 192 300)(2 161 301)(3 162 302)(4 163 303)(5 164 304)(6 165 305)(7 166 306)(8 167 307)(9 168 308)(10 169 309)(11 170 310)(12 171 311)(13 172 312)(14 173 313)(15 174 314)(16 175 315)(17 176 316)(18 177 317)(19 178 318)(20 179 319)(21 180 320)(22 181 289)(23 182 290)(24 183 291)(25 184 292)(26 185 293)(27 186 294)(28 187 295)(29 188 296)(30 189 297)(31 190 298)(32 191 299)(33 414 341)(34 415 342)(35 416 343)(36 385 344)(37 386 345)(38 387 346)(39 388 347)(40 389 348)(41 390 349)(42 391 350)(43 392 351)(44 393 352)(45 394 321)(46 395 322)(47 396 323)(48 397 324)(49 398 325)(50 399 326)(51 400 327)(52 401 328)(53 402 329)(54 403 330)(55 404 331)(56 405 332)(57 406 333)(58 407 334)(59 408 335)(60 409 336)(61 410 337)(62 411 338)(63 412 339)(64 413 340)(65 242 109)(66 243 110)(67 244 111)(68 245 112)(69 246 113)(70 247 114)(71 248 115)(72 249 116)(73 250 117)(74 251 118)(75 252 119)(76 253 120)(77 254 121)(78 255 122)(79 256 123)(80 225 124)(81 226 125)(82 227 126)(83 228 127)(84 229 128)(85 230 97)(86 231 98)(87 232 99)(88 233 100)(89 234 101)(90 235 102)(91 236 103)(92 237 104)(93 238 105)(94 239 106)(95 240 107)(96 241 108)(129 475 275)(130 476 276)(131 477 277)(132 478 278)(133 479 279)(134 480 280)(135 449 281)(136 450 282)(137 451 283)(138 452 284)(139 453 285)(140 454 286)(141 455 287)(142 456 288)(143 457 257)(144 458 258)(145 459 259)(146 460 260)(147 461 261)(148 462 262)(149 463 263)(150 464 264)(151 465 265)(152 466 266)(153 467 267)(154 468 268)(155 469 269)(156 470 270)(157 471 271)(158 472 272)(159 473 273)(160 474 274)(193 439 382)(194 440 383)(195 441 384)(196 442 353)(197 443 354)(198 444 355)(199 445 356)(200 446 357)(201 447 358)(202 448 359)(203 417 360)(204 418 361)(205 419 362)(206 420 363)(207 421 364)(208 422 365)(209 423 366)(210 424 367)(211 425 368)(212 426 369)(213 427 370)(214 428 371)(215 429 372)(216 430 373)(217 431 374)(218 432 375)(219 433 376)(220 434 377)(221 435 378)(222 436 379)(223 437 380)(224 438 381)
(1 209 109 462 341)(2 342 463 110 210)(3 211 111 464 343)(4 344 465 112 212)(5 213 113 466 345)(6 346 467 114 214)(7 215 115 468 347)(8 348 469 116 216)(9 217 117 470 349)(10 350 471 118 218)(11 219 119 472 351)(12 352 473 120 220)(13 221 121 474 321)(14 322 475 122 222)(15 223 123 476 323)(16 324 477 124 224)(17 193 125 478 325)(18 326 479 126 194)(19 195 127 480 327)(20 328 449 128 196)(21 197 97 450 329)(22 330 451 98 198)(23 199 99 452 331)(24 332 453 100 200)(25 201 101 454 333)(26 334 455 102 202)(27 203 103 456 335)(28 336 457 104 204)(29 205 105 458 337)(30 338 459 106 206)(31 207 107 460 339)(32 340 461 108 208)(33 192 423 65 262)(34 263 66 424 161)(35 162 425 67 264)(36 265 68 426 163)(37 164 427 69 266)(38 267 70 428 165)(39 166 429 71 268)(40 269 72 430 167)(41 168 431 73 270)(42 271 74 432 169)(43 170 433 75 272)(44 273 76 434 171)(45 172 435 77 274)(46 275 78 436 173)(47 174 437 79 276)(48 277 80 438 175)(49 176 439 81 278)(50 279 82 440 177)(51 178 441 83 280)(52 281 84 442 179)(53 180 443 85 282)(54 283 86 444 181)(55 182 445 87 284)(56 285 88 446 183)(57 184 447 89 286)(58 287 90 448 185)(59 186 417 91 288)(60 257 92 418 187)(61 188 419 93 258)(62 259 94 420 189)(63 190 421 95 260)(64 261 96 422 191)(129 255 379 313 395)(130 396 314 380 256)(131 225 381 315 397)(132 398 316 382 226)(133 227 383 317 399)(134 400 318 384 228)(135 229 353 319 401)(136 402 320 354 230)(137 231 355 289 403)(138 404 290 356 232)(139 233 357 291 405)(140 406 292 358 234)(141 235 359 293 407)(142 408 294 360 236)(143 237 361 295 409)(144 410 296 362 238)(145 239 363 297 411)(146 412 298 364 240)(147 241 365 299 413)(148 414 300 366 242)(149 243 367 301 415)(150 416 302 368 244)(151 245 369 303 385)(152 386 304 370 246)(153 247 371 305 387)(154 388 306 372 248)(155 249 373 307 389)(156 390 308 374 250)(157 251 375 309 391)(158 392 310 376 252)(159 253 377 311 393)(160 394 312 378 254)
(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)| (1,192,300)(2,161,301)(3,162,302)(4,163,303)(5,164,304)(6,165,305)(7,166,306)(8,167,307)(9,168,308)(10,169,309)(11,170,310)(12,171,311)(13,172,312)(14,173,313)(15,174,314)(16,175,315)(17,176,316)(18,177,317)(19,178,318)(20,179,319)(21,180,320)(22,181,289)(23,182,290)(24,183,291)(25,184,292)(26,185,293)(27,186,294)(28,187,295)(29,188,296)(30,189,297)(31,190,298)(32,191,299)(33,414,341)(34,415,342)(35,416,343)(36,385,344)(37,386,345)(38,387,346)(39,388,347)(40,389,348)(41,390,349)(42,391,350)(43,392,351)(44,393,352)(45,394,321)(46,395,322)(47,396,323)(48,397,324)(49,398,325)(50,399,326)(51,400,327)(52,401,328)(53,402,329)(54,403,330)(55,404,331)(56,405,332)(57,406,333)(58,407,334)(59,408,335)(60,409,336)(61,410,337)(62,411,338)(63,412,339)(64,413,340)(65,242,109)(66,243,110)(67,244,111)(68,245,112)(69,246,113)(70,247,114)(71,248,115)(72,249,116)(73,250,117)(74,251,118)(75,252,119)(76,253,120)(77,254,121)(78,255,122)(79,256,123)(80,225,124)(81,226,125)(82,227,126)(83,228,127)(84,229,128)(85,230,97)(86,231,98)(87,232,99)(88,233,100)(89,234,101)(90,235,102)(91,236,103)(92,237,104)(93,238,105)(94,239,106)(95,240,107)(96,241,108)(129,475,275)(130,476,276)(131,477,277)(132,478,278)(133,479,279)(134,480,280)(135,449,281)(136,450,282)(137,451,283)(138,452,284)(139,453,285)(140,454,286)(141,455,287)(142,456,288)(143,457,257)(144,458,258)(145,459,259)(146,460,260)(147,461,261)(148,462,262)(149,463,263)(150,464,264)(151,465,265)(152,466,266)(153,467,267)(154,468,268)(155,469,269)(156,470,270)(157,471,271)(158,472,272)(159,473,273)(160,474,274)(193,439,382)(194,440,383)(195,441,384)(196,442,353)(197,443,354)(198,444,355)(199,445,356)(200,446,357)(201,447,358)(202,448,359)(203,417,360)(204,418,361)(205,419,362)(206,420,363)(207,421,364)(208,422,365)(209,423,366)(210,424,367)(211,425,368)(212,426,369)(213,427,370)(214,428,371)(215,429,372)(216,430,373)(217,431,374)(218,432,375)(219,433,376)(220,434,377)(221,435,378)(222,436,379)(223,437,380)(224,438,381), (1,209,109,462,341)(2,342,463,110,210)(3,211,111,464,343)(4,344,465,112,212)(5,213,113,466,345)(6,346,467,114,214)(7,215,115,468,347)(8,348,469,116,216)(9,217,117,470,349)(10,350,471,118,218)(11,219,119,472,351)(12,352,473,120,220)(13,221,121,474,321)(14,322,475,122,222)(15,223,123,476,323)(16,324,477,124,224)(17,193,125,478,325)(18,326,479,126,194)(19,195,127,480,327)(20,328,449,128,196)(21,197,97,450,329)(22,330,451,98,198)(23,199,99,452,331)(24,332,453,100,200)(25,201,101,454,333)(26,334,455,102,202)(27,203,103,456,335)(28,336,457,104,204)(29,205,105,458,337)(30,338,459,106,206)(31,207,107,460,339)(32,340,461,108,208)(33,192,423,65,262)(34,263,66,424,161)(35,162,425,67,264)(36,265,68,426,163)(37,164,427,69,266)(38,267,70,428,165)(39,166,429,71,268)(40,269,72,430,167)(41,168,431,73,270)(42,271,74,432,169)(43,170,433,75,272)(44,273,76,434,171)(45,172,435,77,274)(46,275,78,436,173)(47,174,437,79,276)(48,277,80,438,175)(49,176,439,81,278)(50,279,82,440,177)(51,178,441,83,280)(52,281,84,442,179)(53,180,443,85,282)(54,283,86,444,181)(55,182,445,87,284)(56,285,88,446,183)(57,184,447,89,286)(58,287,90,448,185)(59,186,417,91,288)(60,257,92,418,187)(61,188,419,93,258)(62,259,94,420,189)(63,190,421,95,260)(64,261,96,422,191)(129,255,379,313,395)(130,396,314,380,256)(131,225,381,315,397)(132,398,316,382,226)(133,227,383,317,399)(134,400,318,384,228)(135,229,353,319,401)(136,402,320,354,230)(137,231,355,289,403)(138,404,290,356,232)(139,233,357,291,405)(140,406,292,358,234)(141,235,359,293,407)(142,408,294,360,236)(143,237,361,295,409)(144,410,296,362,238)(145,239,363,297,411)(146,412,298,364,240)(147,241,365,299,413)(148,414,300,366,242)(149,243,367,301,415)(150,416,302,368,244)(151,245,369,303,385)(152,386,304,370,246)(153,247,371,305,387)(154,388,306,372,248)(155,249,373,307,389)(156,390,308,374,250)(157,251,375,309,391)(158,392,310,376,252)(159,253,377,311,393)(160,394,312,378,254), (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( (1,192,300)(2,161,301)(3,162,302)(4,163,303)(5,164,304)(6,165,305)(7,166,306)(8,167,307)(9,168,308)(10,169,309)(11,170,310)(12,171,311)(13,172,312)(14,173,313)(15,174,314)(16,175,315)(17,176,316)(18,177,317)(19,178,318)(20,179,319)(21,180,320)(22,181,289)(23,182,290)(24,183,291)(25,184,292)(26,185,293)(27,186,294)(28,187,295)(29,188,296)(30,189,297)(31,190,298)(32,191,299)(33,414,341)(34,415,342)(35,416,343)(36,385,344)(37,386,345)(38,387,346)(39,388,347)(40,389,348)(41,390,349)(42,391,350)(43,392,351)(44,393,352)(45,394,321)(46,395,322)(47,396,323)(48,397,324)(49,398,325)(50,399,326)(51,400,327)(52,401,328)(53,402,329)(54,403,330)(55,404,331)(56,405,332)(57,406,333)(58,407,334)(59,408,335)(60,409,336)(61,410,337)(62,411,338)(63,412,339)(64,413,340)(65,242,109)(66,243,110)(67,244,111)(68,245,112)(69,246,113)(70,247,114)(71,248,115)(72,249,116)(73,250,117)(74,251,118)(75,252,119)(76,253,120)(77,254,121)(78,255,122)(79,256,123)(80,225,124)(81,226,125)(82,227,126)(83,228,127)(84,229,128)(85,230,97)(86,231,98)(87,232,99)(88,233,100)(89,234,101)(90,235,102)(91,236,103)(92,237,104)(93,238,105)(94,239,106)(95,240,107)(96,241,108)(129,475,275)(130,476,276)(131,477,277)(132,478,278)(133,479,279)(134,480,280)(135,449,281)(136,450,282)(137,451,283)(138,452,284)(139,453,285)(140,454,286)(141,455,287)(142,456,288)(143,457,257)(144,458,258)(145,459,259)(146,460,260)(147,461,261)(148,462,262)(149,463,263)(150,464,264)(151,465,265)(152,466,266)(153,467,267)(154,468,268)(155,469,269)(156,470,270)(157,471,271)(158,472,272)(159,473,273)(160,474,274)(193,439,382)(194,440,383)(195,441,384)(196,442,353)(197,443,354)(198,444,355)(199,445,356)(200,446,357)(201,447,358)(202,448,359)(203,417,360)(204,418,361)(205,419,362)(206,420,363)(207,421,364)(208,422,365)(209,423,366)(210,424,367)(211,425,368)(212,426,369)(213,427,370)(214,428,371)(215,429,372)(216,430,373)(217,431,374)(218,432,375)(219,433,376)(220,434,377)(221,435,378)(222,436,379)(223,437,380)(224,438,381), (1,209,109,462,341)(2,342,463,110,210)(3,211,111,464,343)(4,344,465,112,212)(5,213,113,466,345)(6,346,467,114,214)(7,215,115,468,347)(8,348,469,116,216)(9,217,117,470,349)(10,350,471,118,218)(11,219,119,472,351)(12,352,473,120,220)(13,221,121,474,321)(14,322,475,122,222)(15,223,123,476,323)(16,324,477,124,224)(17,193,125,478,325)(18,326,479,126,194)(19,195,127,480,327)(20,328,449,128,196)(21,197,97,450,329)(22,330,451,98,198)(23,199,99,452,331)(24,332,453,100,200)(25,201,101,454,333)(26,334,455,102,202)(27,203,103,456,335)(28,336,457,104,204)(29,205,105,458,337)(30,338,459,106,206)(31,207,107,460,339)(32,340,461,108,208)(33,192,423,65,262)(34,263,66,424,161)(35,162,425,67,264)(36,265,68,426,163)(37,164,427,69,266)(38,267,70,428,165)(39,166,429,71,268)(40,269,72,430,167)(41,168,431,73,270)(42,271,74,432,169)(43,170,433,75,272)(44,273,76,434,171)(45,172,435,77,274)(46,275,78,436,173)(47,174,437,79,276)(48,277,80,438,175)(49,176,439,81,278)(50,279,82,440,177)(51,178,441,83,280)(52,281,84,442,179)(53,180,443,85,282)(54,283,86,444,181)(55,182,445,87,284)(56,285,88,446,183)(57,184,447,89,286)(58,287,90,448,185)(59,186,417,91,288)(60,257,92,418,187)(61,188,419,93,258)(62,259,94,420,189)(63,190,421,95,260)(64,261,96,422,191)(129,255,379,313,395)(130,396,314,380,256)(131,225,381,315,397)(132,398,316,382,226)(133,227,383,317,399)(134,400,318,384,228)(135,229,353,319,401)(136,402,320,354,230)(137,231,355,289,403)(138,404,290,356,232)(139,233,357,291,405)(140,406,292,358,234)(141,235,359,293,407)(142,408,294,360,236)(143,237,361,295,409)(144,410,296,362,238)(145,239,363,297,411)(146,412,298,364,240)(147,241,365,299,413)(148,414,300,366,242)(149,243,367,301,415)(150,416,302,368,244)(151,245,369,303,385)(152,386,304,370,246)(153,247,371,305,387)(154,388,306,372,248)(155,249,373,307,389)(156,390,308,374,250)(157,251,375,309,391)(158,392,310,376,252)(159,253,377,311,393)(160,394,312,378,254), (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([(1,192,300),(2,161,301),(3,162,302),(4,163,303),(5,164,304),(6,165,305),(7,166,306),(8,167,307),(9,168,308),(10,169,309),(11,170,310),(12,171,311),(13,172,312),(14,173,313),(15,174,314),(16,175,315),(17,176,316),(18,177,317),(19,178,318),(20,179,319),(21,180,320),(22,181,289),(23,182,290),(24,183,291),(25,184,292),(26,185,293),(27,186,294),(28,187,295),(29,188,296),(30,189,297),(31,190,298),(32,191,299),(33,414,341),(34,415,342),(35,416,343),(36,385,344),(37,386,345),(38,387,346),(39,388,347),(40,389,348),(41,390,349),(42,391,350),(43,392,351),(44,393,352),(45,394,321),(46,395,322),(47,396,323),(48,397,324),(49,398,325),(50,399,326),(51,400,327),(52,401,328),(53,402,329),(54,403,330),(55,404,331),(56,405,332),(57,406,333),(58,407,334),(59,408,335),(60,409,336),(61,410,337),(62,411,338),(63,412,339),(64,413,340),(65,242,109),(66,243,110),(67,244,111),(68,245,112),(69,246,113),(70,247,114),(71,248,115),(72,249,116),(73,250,117),(74,251,118),(75,252,119),(76,253,120),(77,254,121),(78,255,122),(79,256,123),(80,225,124),(81,226,125),(82,227,126),(83,228,127),(84,229,128),(85,230,97),(86,231,98),(87,232,99),(88,233,100),(89,234,101),(90,235,102),(91,236,103),(92,237,104),(93,238,105),(94,239,106),(95,240,107),(96,241,108),(129,475,275),(130,476,276),(131,477,277),(132,478,278),(133,479,279),(134,480,280),(135,449,281),(136,450,282),(137,451,283),(138,452,284),(139,453,285),(140,454,286),(141,455,287),(142,456,288),(143,457,257),(144,458,258),(145,459,259),(146,460,260),(147,461,261),(148,462,262),(149,463,263),(150,464,264),(151,465,265),(152,466,266),(153,467,267),(154,468,268),(155,469,269),(156,470,270),(157,471,271),(158,472,272),(159,473,273),(160,474,274),(193,439,382),(194,440,383),(195,441,384),(196,442,353),(197,443,354),(198,444,355),(199,445,356),(200,446,357),(201,447,358),(202,448,359),(203,417,360),(204,418,361),(205,419,362),(206,420,363),(207,421,364),(208,422,365),(209,423,366),(210,424,367),(211,425,368),(212,426,369),(213,427,370),(214,428,371),(215,429,372),(216,430,373),(217,431,374),(218,432,375),(219,433,376),(220,434,377),(221,435,378),(222,436,379),(223,437,380),(224,438,381)], [(1,209,109,462,341),(2,342,463,110,210),(3,211,111,464,343),(4,344,465,112,212),(5,213,113,466,345),(6,346,467,114,214),(7,215,115,468,347),(8,348,469,116,216),(9,217,117,470,349),(10,350,471,118,218),(11,219,119,472,351),(12,352,473,120,220),(13,221,121,474,321),(14,322,475,122,222),(15,223,123,476,323),(16,324,477,124,224),(17,193,125,478,325),(18,326,479,126,194),(19,195,127,480,327),(20,328,449,128,196),(21,197,97,450,329),(22,330,451,98,198),(23,199,99,452,331),(24,332,453,100,200),(25,201,101,454,333),(26,334,455,102,202),(27,203,103,456,335),(28,336,457,104,204),(29,205,105,458,337),(30,338,459,106,206),(31,207,107,460,339),(32,340,461,108,208),(33,192,423,65,262),(34,263,66,424,161),(35,162,425,67,264),(36,265,68,426,163),(37,164,427,69,266),(38,267,70,428,165),(39,166,429,71,268),(40,269,72,430,167),(41,168,431,73,270),(42,271,74,432,169),(43,170,433,75,272),(44,273,76,434,171),(45,172,435,77,274),(46,275,78,436,173),(47,174,437,79,276),(48,277,80,438,175),(49,176,439,81,278),(50,279,82,440,177),(51,178,441,83,280),(52,281,84,442,179),(53,180,443,85,282),(54,283,86,444,181),(55,182,445,87,284),(56,285,88,446,183),(57,184,447,89,286),(58,287,90,448,185),(59,186,417,91,288),(60,257,92,418,187),(61,188,419,93,258),(62,259,94,420,189),(63,190,421,95,260),(64,261,96,422,191),(129,255,379,313,395),(130,396,314,380,256),(131,225,381,315,397),(132,398,316,382,226),(133,227,383,317,399),(134,400,318,384,228),(135,229,353,319,401),(136,402,320,354,230),(137,231,355,289,403),(138,404,290,356,232),(139,233,357,291,405),(140,406,292,358,234),(141,235,359,293,407),(142,408,294,360,236),(143,237,361,295,409),(144,410,296,362,238),(145,239,363,297,411),(146,412,298,364,240),(147,241,365,299,413),(148,414,300,366,242),(149,243,367,301,415),(150,416,302,368,244),(151,245,369,303,385),(152,386,304,370,246),(153,247,371,305,387),(154,388,306,372,248),(155,249,373,307,389),(156,390,308,374,250),(157,251,375,309,391),(158,392,310,376,252),(159,253,377,311,393),(160,394,312,378,254)], [(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)])

192 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 6A 6B 8A 8B 8C 8D 10A 10B 12A 12B 12C 12D 15A 15B 15C 15D 16A ··· 16H 20A 20B 20C 20D 24A ··· 24H 30A 30B 30C 30D 32A ··· 32P 40A ··· 40H 48A ··· 48P 60A ··· 60H 80A ··· 80P 96A ··· 96AF 120A ··· 120P 240A ··· 240AF order 1 2 3 3 4 4 5 5 6 6 8 8 8 8 10 10 12 12 12 12 15 15 15 15 16 ··· 16 20 20 20 20 24 ··· 24 30 30 30 30 32 ··· 32 40 ··· 40 48 ··· 48 60 ··· 60 80 ··· 80 96 ··· 96 120 ··· 120 240 ··· 240 size 1 1 1 1 1 1 2 2 1 1 1 1 1 1 2 2 1 1 1 1 2 2 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 5 ··· 5 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 5 ··· 5 2 ··· 2 2 ··· 2

192 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C8 C12 C16 C24 C32 C48 C96 D5 Dic5 C3×D5 C5⋊2C8 C3×Dic5 C5⋊2C16 C3×C5⋊2C8 C5⋊2C32 C3×C5⋊2C16 C3×C5⋊2C32 kernel C3×C5⋊2C32 C240 C5⋊2C32 C120 C80 C60 C40 C30 C20 C15 C10 C5 C48 C24 C16 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 4 4 8 8 16 16 32 2 2 4 4 4 8 8 16 16 32

Matrix representation of C3×C52C32 in GL3(𝔽3361) generated by

 892 0 0 0 892 0 0 0 892
,
 1 0 0 0 489 3360 0 490 3360
,
 1031 0 0 0 2106 286 0 2452 1255
G:=sub<GL(3,GF(3361))| [892,0,0,0,892,0,0,0,892],[1,0,0,0,489,490,0,3360,3360],[1031,0,0,0,2106,2452,0,286,1255] >;

C3×C52C32 in GAP, Magma, Sage, TeX

C_3\times C_5\rtimes_2C_{32}
% in TeX

G:=Group("C3xC5:2C32");
// GroupNames label

G:=SmallGroup(480,2);
// by ID

G=gap.SmallGroup(480,2);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,-2,-5,42,58,80,102,18822]);
// 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^-1>;
// generators/relations

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