direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C3⋊Q32, C15⋊9Q32, C40.59D6, C30.56D8, C60.139D4, C120.66C22, Dic12.2C10, C3⋊C16.C10, C3⋊2(C5×Q32), Q16.(C5×S3), C8.7(S3×C10), C6.11(C5×D8), C12.6(C5×D4), C24.5(C2×C10), (C5×Q16).2S3, (C15×Q16).3C2, (C3×Q16).1C10, C10.27(D4⋊S3), C20.69(C3⋊D4), (C5×Dic12).4C2, (C5×C3⋊C16).2C2, C2.7(C5×D4⋊S3), C4.4(C5×C3⋊D4), SmallGroup(480,148)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C3⋊Q32
G = < a,b,c,d | a5=b3=c16=1, d2=c8, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Smallest permutation representation of C5×C3⋊Q32
►Regular action on 480 points
Generators in S480
(1 21 195 244 296)(2 22 196 245 297)(3 23 197 246 298)(4 24 198 247 299)(5 25 199 248 300)(6 26 200 249 301)(7 27 201 250 302)(8 28 202 251 303)(9 29 203 252 304)(10 30 204 253 289)(11 31 205 254 290)(12 32 206 255 291)(13 17 207 256 292)(14 18 208 241 293)(15 19 193 242 294)(16 20 194 243 295)(33 367 456 390 230)(34 368 457 391 231)(35 353 458 392 232)(36 354 459 393 233)(37 355 460 394 234)(38 356 461 395 235)(39 357 462 396 236)(40 358 463 397 237)(41 359 464 398 238)(42 360 449 399 239)(43 361 450 400 240)(44 362 451 385 225)(45 363 452 386 226)(46 364 453 387 227)(47 365 454 388 228)(48 366 455 389 229)(49 381 444 144 213)(50 382 445 129 214)(51 383 446 130 215)(52 384 447 131 216)(53 369 448 132 217)(54 370 433 133 218)(55 371 434 134 219)(56 372 435 135 220)(57 373 436 136 221)(58 374 437 137 222)(59 375 438 138 223)(60 376 439 139 224)(61 377 440 140 209)(62 378 441 141 210)(63 379 442 142 211)(64 380 443 143 212)(65 161 150 309 471)(66 162 151 310 472)(67 163 152 311 473)(68 164 153 312 474)(69 165 154 313 475)(70 166 155 314 476)(71 167 156 315 477)(72 168 157 316 478)(73 169 158 317 479)(74 170 159 318 480)(75 171 160 319 465)(76 172 145 320 466)(77 173 146 305 467)(78 174 147 306 468)(79 175 148 307 469)(80 176 149 308 470)(81 429 265 128 349)(82 430 266 113 350)(83 431 267 114 351)(84 432 268 115 352)(85 417 269 116 337)(86 418 270 117 338)(87 419 271 118 339)(88 420 272 119 340)(89 421 257 120 341)(90 422 258 121 342)(91 423 259 122 343)(92 424 260 123 344)(93 425 261 124 345)(94 426 262 125 346)(95 427 263 126 347)(96 428 264 127 348)(97 321 278 178 414)(98 322 279 179 415)(99 323 280 180 416)(100 324 281 181 401)(101 325 282 182 402)(102 326 283 183 403)(103 327 284 184 404)(104 328 285 185 405)(105 329 286 186 406)(106 330 287 187 407)(107 331 288 188 408)(108 332 273 189 409)(109 333 274 190 410)(110 334 275 191 411)(111 335 276 192 412)(112 336 277 177 413)
(1 422 310)(2 311 423)(3 424 312)(4 313 425)(5 426 314)(6 315 427)(7 428 316)(8 317 429)(9 430 318)(10 319 431)(11 432 320)(12 305 417)(13 418 306)(14 307 419)(15 420 308)(16 309 421)(17 270 468)(18 469 271)(19 272 470)(20 471 257)(21 258 472)(22 473 259)(23 260 474)(24 475 261)(25 262 476)(26 477 263)(27 264 478)(28 479 265)(29 266 480)(30 465 267)(31 268 466)(32 467 269)(33 407 439)(34 440 408)(35 409 441)(36 442 410)(37 411 443)(38 444 412)(39 413 445)(40 446 414)(41 415 447)(42 448 416)(43 401 433)(44 434 402)(45 403 435)(46 436 404)(47 405 437)(48 438 406)(49 276 395)(50 396 277)(51 278 397)(52 398 279)(53 280 399)(54 400 281)(55 282 385)(56 386 283)(57 284 387)(58 388 285)(59 286 389)(60 390 287)(61 288 391)(62 392 273)(63 274 393)(64 394 275)(65 120 194)(66 195 121)(67 122 196)(68 197 123)(69 124 198)(70 199 125)(71 126 200)(72 201 127)(73 128 202)(74 203 113)(75 114 204)(76 205 115)(77 116 206)(78 207 117)(79 118 208)(80 193 119)(81 303 158)(82 159 304)(83 289 160)(84 145 290)(85 291 146)(86 147 292)(87 293 148)(88 149 294)(89 295 150)(90 151 296)(91 297 152)(92 153 298)(93 299 154)(94 155 300)(95 301 156)(96 157 302)(97 358 130)(98 131 359)(99 360 132)(100 133 361)(101 362 134)(102 135 363)(103 364 136)(104 137 365)(105 366 138)(106 139 367)(107 368 140)(108 141 353)(109 354 142)(110 143 355)(111 356 144)(112 129 357)(161 341 243)(162 244 342)(163 343 245)(164 246 344)(165 345 247)(166 248 346)(167 347 249)(168 250 348)(169 349 251)(170 252 350)(171 351 253)(172 254 352)(173 337 255)(174 256 338)(175 339 241)(176 242 340)(177 382 236)(178 237 383)(179 384 238)(180 239 369)(181 370 240)(182 225 371)(183 372 226)(184 227 373)(185 374 228)(186 229 375)(187 376 230)(188 231 377)(189 378 232)(190 233 379)(191 380 234)(192 235 381)(209 331 457)(210 458 332)(211 333 459)(212 460 334)(213 335 461)(214 462 336)(215 321 463)(216 464 322)(217 323 449)(218 450 324)(219 325 451)(220 452 326)(221 327 453)(222 454 328)(223 329 455)(224 456 330)
(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)
(1 280 9 288)(2 279 10 287)(3 278 11 286)(4 277 12 285)(5 276 13 284)(6 275 14 283)(7 274 15 282)(8 273 16 281)(17 184 25 192)(18 183 26 191)(19 182 27 190)(20 181 28 189)(21 180 29 188)(22 179 30 187)(23 178 31 186)(24 177 32 185)(33 122 41 114)(34 121 42 113)(35 120 43 128)(36 119 44 127)(37 118 45 126)(38 117 46 125)(39 116 47 124)(40 115 48 123)(49 306 57 314)(50 305 58 313)(51 320 59 312)(52 319 60 311)(53 318 61 310)(54 317 62 309)(55 316 63 308)(56 315 64 307)(65 433 73 441)(66 448 74 440)(67 447 75 439)(68 446 76 438)(69 445 77 437)(70 444 78 436)(71 443 79 435)(72 442 80 434)(81 458 89 450)(82 457 90 449)(83 456 91 464)(84 455 92 463)(85 454 93 462)(86 453 94 461)(87 452 95 460)(88 451 96 459)(97 254 105 246)(98 253 106 245)(99 252 107 244)(100 251 108 243)(101 250 109 242)(102 249 110 241)(103 248 111 256)(104 247 112 255)(129 173 137 165)(130 172 138 164)(131 171 139 163)(132 170 140 162)(133 169 141 161)(134 168 142 176)(135 167 143 175)(136 166 144 174)(145 223 153 215)(146 222 154 214)(147 221 155 213)(148 220 156 212)(149 219 157 211)(150 218 158 210)(151 217 159 209)(152 216 160 224)(193 402 201 410)(194 401 202 409)(195 416 203 408)(196 415 204 407)(197 414 205 406)(198 413 206 405)(199 412 207 404)(200 411 208 403)(225 264 233 272)(226 263 234 271)(227 262 235 270)(228 261 236 269)(229 260 237 268)(230 259 238 267)(231 258 239 266)(232 257 240 265)(289 330 297 322)(290 329 298 321)(291 328 299 336)(292 327 300 335)(293 326 301 334)(294 325 302 333)(295 324 303 332)(296 323 304 331)(337 365 345 357)(338 364 346 356)(339 363 347 355)(340 362 348 354)(341 361 349 353)(342 360 350 368)(343 359 351 367)(344 358 352 366)(369 480 377 472)(370 479 378 471)(371 478 379 470)(372 477 380 469)(373 476 381 468)(374 475 382 467)(375 474 383 466)(376 473 384 465)(385 428 393 420)(386 427 394 419)(387 426 395 418)(388 425 396 417)(389 424 397 432)(390 423 398 431)(391 422 399 430)(392 421 400 429)
G:=sub<Sym(480)| (1,21,195,244,296)(2,22,196,245,297)(3,23,197,246,298)(4,24,198,247,299)(5,25,199,248,300)(6,26,200,249,301)(7,27,201,250,302)(8,28,202,251,303)(9,29,203,252,304)(10,30,204,253,289)(11,31,205,254,290)(12,32,206,255,291)(13,17,207,256,292)(14,18,208,241,293)(15,19,193,242,294)(16,20,194,243,295)(33,367,456,390,230)(34,368,457,391,231)(35,353,458,392,232)(36,354,459,393,233)(37,355,460,394,234)(38,356,461,395,235)(39,357,462,396,236)(40,358,463,397,237)(41,359,464,398,238)(42,360,449,399,239)(43,361,450,400,240)(44,362,451,385,225)(45,363,452,386,226)(46,364,453,387,227)(47,365,454,388,228)(48,366,455,389,229)(49,381,444,144,213)(50,382,445,129,214)(51,383,446,130,215)(52,384,447,131,216)(53,369,448,132,217)(54,370,433,133,218)(55,371,434,134,219)(56,372,435,135,220)(57,373,436,136,221)(58,374,437,137,222)(59,375,438,138,223)(60,376,439,139,224)(61,377,440,140,209)(62,378,441,141,210)(63,379,442,142,211)(64,380,443,143,212)(65,161,150,309,471)(66,162,151,310,472)(67,163,152,311,473)(68,164,153,312,474)(69,165,154,313,475)(70,166,155,314,476)(71,167,156,315,477)(72,168,157,316,478)(73,169,158,317,479)(74,170,159,318,480)(75,171,160,319,465)(76,172,145,320,466)(77,173,146,305,467)(78,174,147,306,468)(79,175,148,307,469)(80,176,149,308,470)(81,429,265,128,349)(82,430,266,113,350)(83,431,267,114,351)(84,432,268,115,352)(85,417,269,116,337)(86,418,270,117,338)(87,419,271,118,339)(88,420,272,119,340)(89,421,257,120,341)(90,422,258,121,342)(91,423,259,122,343)(92,424,260,123,344)(93,425,261,124,345)(94,426,262,125,346)(95,427,263,126,347)(96,428,264,127,348)(97,321,278,178,414)(98,322,279,179,415)(99,323,280,180,416)(100,324,281,181,401)(101,325,282,182,402)(102,326,283,183,403)(103,327,284,184,404)(104,328,285,185,405)(105,329,286,186,406)(106,330,287,187,407)(107,331,288,188,408)(108,332,273,189,409)(109,333,274,190,410)(110,334,275,191,411)(111,335,276,192,412)(112,336,277,177,413), (1,422,310)(2,311,423)(3,424,312)(4,313,425)(5,426,314)(6,315,427)(7,428,316)(8,317,429)(9,430,318)(10,319,431)(11,432,320)(12,305,417)(13,418,306)(14,307,419)(15,420,308)(16,309,421)(17,270,468)(18,469,271)(19,272,470)(20,471,257)(21,258,472)(22,473,259)(23,260,474)(24,475,261)(25,262,476)(26,477,263)(27,264,478)(28,479,265)(29,266,480)(30,465,267)(31,268,466)(32,467,269)(33,407,439)(34,440,408)(35,409,441)(36,442,410)(37,411,443)(38,444,412)(39,413,445)(40,446,414)(41,415,447)(42,448,416)(43,401,433)(44,434,402)(45,403,435)(46,436,404)(47,405,437)(48,438,406)(49,276,395)(50,396,277)(51,278,397)(52,398,279)(53,280,399)(54,400,281)(55,282,385)(56,386,283)(57,284,387)(58,388,285)(59,286,389)(60,390,287)(61,288,391)(62,392,273)(63,274,393)(64,394,275)(65,120,194)(66,195,121)(67,122,196)(68,197,123)(69,124,198)(70,199,125)(71,126,200)(72,201,127)(73,128,202)(74,203,113)(75,114,204)(76,205,115)(77,116,206)(78,207,117)(79,118,208)(80,193,119)(81,303,158)(82,159,304)(83,289,160)(84,145,290)(85,291,146)(86,147,292)(87,293,148)(88,149,294)(89,295,150)(90,151,296)(91,297,152)(92,153,298)(93,299,154)(94,155,300)(95,301,156)(96,157,302)(97,358,130)(98,131,359)(99,360,132)(100,133,361)(101,362,134)(102,135,363)(103,364,136)(104,137,365)(105,366,138)(106,139,367)(107,368,140)(108,141,353)(109,354,142)(110,143,355)(111,356,144)(112,129,357)(161,341,243)(162,244,342)(163,343,245)(164,246,344)(165,345,247)(166,248,346)(167,347,249)(168,250,348)(169,349,251)(170,252,350)(171,351,253)(172,254,352)(173,337,255)(174,256,338)(175,339,241)(176,242,340)(177,382,236)(178,237,383)(179,384,238)(180,239,369)(181,370,240)(182,225,371)(183,372,226)(184,227,373)(185,374,228)(186,229,375)(187,376,230)(188,231,377)(189,378,232)(190,233,379)(191,380,234)(192,235,381)(209,331,457)(210,458,332)(211,333,459)(212,460,334)(213,335,461)(214,462,336)(215,321,463)(216,464,322)(217,323,449)(218,450,324)(219,325,451)(220,452,326)(221,327,453)(222,454,328)(223,329,455)(224,456,330), (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), (1,280,9,288)(2,279,10,287)(3,278,11,286)(4,277,12,285)(5,276,13,284)(6,275,14,283)(7,274,15,282)(8,273,16,281)(17,184,25,192)(18,183,26,191)(19,182,27,190)(20,181,28,189)(21,180,29,188)(22,179,30,187)(23,178,31,186)(24,177,32,185)(33,122,41,114)(34,121,42,113)(35,120,43,128)(36,119,44,127)(37,118,45,126)(38,117,46,125)(39,116,47,124)(40,115,48,123)(49,306,57,314)(50,305,58,313)(51,320,59,312)(52,319,60,311)(53,318,61,310)(54,317,62,309)(55,316,63,308)(56,315,64,307)(65,433,73,441)(66,448,74,440)(67,447,75,439)(68,446,76,438)(69,445,77,437)(70,444,78,436)(71,443,79,435)(72,442,80,434)(81,458,89,450)(82,457,90,449)(83,456,91,464)(84,455,92,463)(85,454,93,462)(86,453,94,461)(87,452,95,460)(88,451,96,459)(97,254,105,246)(98,253,106,245)(99,252,107,244)(100,251,108,243)(101,250,109,242)(102,249,110,241)(103,248,111,256)(104,247,112,255)(129,173,137,165)(130,172,138,164)(131,171,139,163)(132,170,140,162)(133,169,141,161)(134,168,142,176)(135,167,143,175)(136,166,144,174)(145,223,153,215)(146,222,154,214)(147,221,155,213)(148,220,156,212)(149,219,157,211)(150,218,158,210)(151,217,159,209)(152,216,160,224)(193,402,201,410)(194,401,202,409)(195,416,203,408)(196,415,204,407)(197,414,205,406)(198,413,206,405)(199,412,207,404)(200,411,208,403)(225,264,233,272)(226,263,234,271)(227,262,235,270)(228,261,236,269)(229,260,237,268)(230,259,238,267)(231,258,239,266)(232,257,240,265)(289,330,297,322)(290,329,298,321)(291,328,299,336)(292,327,300,335)(293,326,301,334)(294,325,302,333)(295,324,303,332)(296,323,304,331)(337,365,345,357)(338,364,346,356)(339,363,347,355)(340,362,348,354)(341,361,349,353)(342,360,350,368)(343,359,351,367)(344,358,352,366)(369,480,377,472)(370,479,378,471)(371,478,379,470)(372,477,380,469)(373,476,381,468)(374,475,382,467)(375,474,383,466)(376,473,384,465)(385,428,393,420)(386,427,394,419)(387,426,395,418)(388,425,396,417)(389,424,397,432)(390,423,398,431)(391,422,399,430)(392,421,400,429)>;
G:=Group( (1,21,195,244,296)(2,22,196,245,297)(3,23,197,246,298)(4,24,198,247,299)(5,25,199,248,300)(6,26,200,249,301)(7,27,201,250,302)(8,28,202,251,303)(9,29,203,252,304)(10,30,204,253,289)(11,31,205,254,290)(12,32,206,255,291)(13,17,207,256,292)(14,18,208,241,293)(15,19,193,242,294)(16,20,194,243,295)(33,367,456,390,230)(34,368,457,391,231)(35,353,458,392,232)(36,354,459,393,233)(37,355,460,394,234)(38,356,461,395,235)(39,357,462,396,236)(40,358,463,397,237)(41,359,464,398,238)(42,360,449,399,239)(43,361,450,400,240)(44,362,451,385,225)(45,363,452,386,226)(46,364,453,387,227)(47,365,454,388,228)(48,366,455,389,229)(49,381,444,144,213)(50,382,445,129,214)(51,383,446,130,215)(52,384,447,131,216)(53,369,448,132,217)(54,370,433,133,218)(55,371,434,134,219)(56,372,435,135,220)(57,373,436,136,221)(58,374,437,137,222)(59,375,438,138,223)(60,376,439,139,224)(61,377,440,140,209)(62,378,441,141,210)(63,379,442,142,211)(64,380,443,143,212)(65,161,150,309,471)(66,162,151,310,472)(67,163,152,311,473)(68,164,153,312,474)(69,165,154,313,475)(70,166,155,314,476)(71,167,156,315,477)(72,168,157,316,478)(73,169,158,317,479)(74,170,159,318,480)(75,171,160,319,465)(76,172,145,320,466)(77,173,146,305,467)(78,174,147,306,468)(79,175,148,307,469)(80,176,149,308,470)(81,429,265,128,349)(82,430,266,113,350)(83,431,267,114,351)(84,432,268,115,352)(85,417,269,116,337)(86,418,270,117,338)(87,419,271,118,339)(88,420,272,119,340)(89,421,257,120,341)(90,422,258,121,342)(91,423,259,122,343)(92,424,260,123,344)(93,425,261,124,345)(94,426,262,125,346)(95,427,263,126,347)(96,428,264,127,348)(97,321,278,178,414)(98,322,279,179,415)(99,323,280,180,416)(100,324,281,181,401)(101,325,282,182,402)(102,326,283,183,403)(103,327,284,184,404)(104,328,285,185,405)(105,329,286,186,406)(106,330,287,187,407)(107,331,288,188,408)(108,332,273,189,409)(109,333,274,190,410)(110,334,275,191,411)(111,335,276,192,412)(112,336,277,177,413), (1,422,310)(2,311,423)(3,424,312)(4,313,425)(5,426,314)(6,315,427)(7,428,316)(8,317,429)(9,430,318)(10,319,431)(11,432,320)(12,305,417)(13,418,306)(14,307,419)(15,420,308)(16,309,421)(17,270,468)(18,469,271)(19,272,470)(20,471,257)(21,258,472)(22,473,259)(23,260,474)(24,475,261)(25,262,476)(26,477,263)(27,264,478)(28,479,265)(29,266,480)(30,465,267)(31,268,466)(32,467,269)(33,407,439)(34,440,408)(35,409,441)(36,442,410)(37,411,443)(38,444,412)(39,413,445)(40,446,414)(41,415,447)(42,448,416)(43,401,433)(44,434,402)(45,403,435)(46,436,404)(47,405,437)(48,438,406)(49,276,395)(50,396,277)(51,278,397)(52,398,279)(53,280,399)(54,400,281)(55,282,385)(56,386,283)(57,284,387)(58,388,285)(59,286,389)(60,390,287)(61,288,391)(62,392,273)(63,274,393)(64,394,275)(65,120,194)(66,195,121)(67,122,196)(68,197,123)(69,124,198)(70,199,125)(71,126,200)(72,201,127)(73,128,202)(74,203,113)(75,114,204)(76,205,115)(77,116,206)(78,207,117)(79,118,208)(80,193,119)(81,303,158)(82,159,304)(83,289,160)(84,145,290)(85,291,146)(86,147,292)(87,293,148)(88,149,294)(89,295,150)(90,151,296)(91,297,152)(92,153,298)(93,299,154)(94,155,300)(95,301,156)(96,157,302)(97,358,130)(98,131,359)(99,360,132)(100,133,361)(101,362,134)(102,135,363)(103,364,136)(104,137,365)(105,366,138)(106,139,367)(107,368,140)(108,141,353)(109,354,142)(110,143,355)(111,356,144)(112,129,357)(161,341,243)(162,244,342)(163,343,245)(164,246,344)(165,345,247)(166,248,346)(167,347,249)(168,250,348)(169,349,251)(170,252,350)(171,351,253)(172,254,352)(173,337,255)(174,256,338)(175,339,241)(176,242,340)(177,382,236)(178,237,383)(179,384,238)(180,239,369)(181,370,240)(182,225,371)(183,372,226)(184,227,373)(185,374,228)(186,229,375)(187,376,230)(188,231,377)(189,378,232)(190,233,379)(191,380,234)(192,235,381)(209,331,457)(210,458,332)(211,333,459)(212,460,334)(213,335,461)(214,462,336)(215,321,463)(216,464,322)(217,323,449)(218,450,324)(219,325,451)(220,452,326)(221,327,453)(222,454,328)(223,329,455)(224,456,330), (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), (1,280,9,288)(2,279,10,287)(3,278,11,286)(4,277,12,285)(5,276,13,284)(6,275,14,283)(7,274,15,282)(8,273,16,281)(17,184,25,192)(18,183,26,191)(19,182,27,190)(20,181,28,189)(21,180,29,188)(22,179,30,187)(23,178,31,186)(24,177,32,185)(33,122,41,114)(34,121,42,113)(35,120,43,128)(36,119,44,127)(37,118,45,126)(38,117,46,125)(39,116,47,124)(40,115,48,123)(49,306,57,314)(50,305,58,313)(51,320,59,312)(52,319,60,311)(53,318,61,310)(54,317,62,309)(55,316,63,308)(56,315,64,307)(65,433,73,441)(66,448,74,440)(67,447,75,439)(68,446,76,438)(69,445,77,437)(70,444,78,436)(71,443,79,435)(72,442,80,434)(81,458,89,450)(82,457,90,449)(83,456,91,464)(84,455,92,463)(85,454,93,462)(86,453,94,461)(87,452,95,460)(88,451,96,459)(97,254,105,246)(98,253,106,245)(99,252,107,244)(100,251,108,243)(101,250,109,242)(102,249,110,241)(103,248,111,256)(104,247,112,255)(129,173,137,165)(130,172,138,164)(131,171,139,163)(132,170,140,162)(133,169,141,161)(134,168,142,176)(135,167,143,175)(136,166,144,174)(145,223,153,215)(146,222,154,214)(147,221,155,213)(148,220,156,212)(149,219,157,211)(150,218,158,210)(151,217,159,209)(152,216,160,224)(193,402,201,410)(194,401,202,409)(195,416,203,408)(196,415,204,407)(197,414,205,406)(198,413,206,405)(199,412,207,404)(200,411,208,403)(225,264,233,272)(226,263,234,271)(227,262,235,270)(228,261,236,269)(229,260,237,268)(230,259,238,267)(231,258,239,266)(232,257,240,265)(289,330,297,322)(290,329,298,321)(291,328,299,336)(292,327,300,335)(293,326,301,334)(294,325,302,333)(295,324,303,332)(296,323,304,331)(337,365,345,357)(338,364,346,356)(339,363,347,355)(340,362,348,354)(341,361,349,353)(342,360,350,368)(343,359,351,367)(344,358,352,366)(369,480,377,472)(370,479,378,471)(371,478,379,470)(372,477,380,469)(373,476,381,468)(374,475,382,467)(375,474,383,466)(376,473,384,465)(385,428,393,420)(386,427,394,419)(387,426,395,418)(388,425,396,417)(389,424,397,432)(390,423,398,431)(391,422,399,430)(392,421,400,429) );
G=PermutationGroup([[(1,21,195,244,296),(2,22,196,245,297),(3,23,197,246,298),(4,24,198,247,299),(5,25,199,248,300),(6,26,200,249,301),(7,27,201,250,302),(8,28,202,251,303),(9,29,203,252,304),(10,30,204,253,289),(11,31,205,254,290),(12,32,206,255,291),(13,17,207,256,292),(14,18,208,241,293),(15,19,193,242,294),(16,20,194,243,295),(33,367,456,390,230),(34,368,457,391,231),(35,353,458,392,232),(36,354,459,393,233),(37,355,460,394,234),(38,356,461,395,235),(39,357,462,396,236),(40,358,463,397,237),(41,359,464,398,238),(42,360,449,399,239),(43,361,450,400,240),(44,362,451,385,225),(45,363,452,386,226),(46,364,453,387,227),(47,365,454,388,228),(48,366,455,389,229),(49,381,444,144,213),(50,382,445,129,214),(51,383,446,130,215),(52,384,447,131,216),(53,369,448,132,217),(54,370,433,133,218),(55,371,434,134,219),(56,372,435,135,220),(57,373,436,136,221),(58,374,437,137,222),(59,375,438,138,223),(60,376,439,139,224),(61,377,440,140,209),(62,378,441,141,210),(63,379,442,142,211),(64,380,443,143,212),(65,161,150,309,471),(66,162,151,310,472),(67,163,152,311,473),(68,164,153,312,474),(69,165,154,313,475),(70,166,155,314,476),(71,167,156,315,477),(72,168,157,316,478),(73,169,158,317,479),(74,170,159,318,480),(75,171,160,319,465),(76,172,145,320,466),(77,173,146,305,467),(78,174,147,306,468),(79,175,148,307,469),(80,176,149,308,470),(81,429,265,128,349),(82,430,266,113,350),(83,431,267,114,351),(84,432,268,115,352),(85,417,269,116,337),(86,418,270,117,338),(87,419,271,118,339),(88,420,272,119,340),(89,421,257,120,341),(90,422,258,121,342),(91,423,259,122,343),(92,424,260,123,344),(93,425,261,124,345),(94,426,262,125,346),(95,427,263,126,347),(96,428,264,127,348),(97,321,278,178,414),(98,322,279,179,415),(99,323,280,180,416),(100,324,281,181,401),(101,325,282,182,402),(102,326,283,183,403),(103,327,284,184,404),(104,328,285,185,405),(105,329,286,186,406),(106,330,287,187,407),(107,331,288,188,408),(108,332,273,189,409),(109,333,274,190,410),(110,334,275,191,411),(111,335,276,192,412),(112,336,277,177,413)], [(1,422,310),(2,311,423),(3,424,312),(4,313,425),(5,426,314),(6,315,427),(7,428,316),(8,317,429),(9,430,318),(10,319,431),(11,432,320),(12,305,417),(13,418,306),(14,307,419),(15,420,308),(16,309,421),(17,270,468),(18,469,271),(19,272,470),(20,471,257),(21,258,472),(22,473,259),(23,260,474),(24,475,261),(25,262,476),(26,477,263),(27,264,478),(28,479,265),(29,266,480),(30,465,267),(31,268,466),(32,467,269),(33,407,439),(34,440,408),(35,409,441),(36,442,410),(37,411,443),(38,444,412),(39,413,445),(40,446,414),(41,415,447),(42,448,416),(43,401,433),(44,434,402),(45,403,435),(46,436,404),(47,405,437),(48,438,406),(49,276,395),(50,396,277),(51,278,397),(52,398,279),(53,280,399),(54,400,281),(55,282,385),(56,386,283),(57,284,387),(58,388,285),(59,286,389),(60,390,287),(61,288,391),(62,392,273),(63,274,393),(64,394,275),(65,120,194),(66,195,121),(67,122,196),(68,197,123),(69,124,198),(70,199,125),(71,126,200),(72,201,127),(73,128,202),(74,203,113),(75,114,204),(76,205,115),(77,116,206),(78,207,117),(79,118,208),(80,193,119),(81,303,158),(82,159,304),(83,289,160),(84,145,290),(85,291,146),(86,147,292),(87,293,148),(88,149,294),(89,295,150),(90,151,296),(91,297,152),(92,153,298),(93,299,154),(94,155,300),(95,301,156),(96,157,302),(97,358,130),(98,131,359),(99,360,132),(100,133,361),(101,362,134),(102,135,363),(103,364,136),(104,137,365),(105,366,138),(106,139,367),(107,368,140),(108,141,353),(109,354,142),(110,143,355),(111,356,144),(112,129,357),(161,341,243),(162,244,342),(163,343,245),(164,246,344),(165,345,247),(166,248,346),(167,347,249),(168,250,348),(169,349,251),(170,252,350),(171,351,253),(172,254,352),(173,337,255),(174,256,338),(175,339,241),(176,242,340),(177,382,236),(178,237,383),(179,384,238),(180,239,369),(181,370,240),(182,225,371),(183,372,226),(184,227,373),(185,374,228),(186,229,375),(187,376,230),(188,231,377),(189,378,232),(190,233,379),(191,380,234),(192,235,381),(209,331,457),(210,458,332),(211,333,459),(212,460,334),(213,335,461),(214,462,336),(215,321,463),(216,464,322),(217,323,449),(218,450,324),(219,325,451),(220,452,326),(221,327,453),(222,454,328),(223,329,455),(224,456,330)], [(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)], [(1,280,9,288),(2,279,10,287),(3,278,11,286),(4,277,12,285),(5,276,13,284),(6,275,14,283),(7,274,15,282),(8,273,16,281),(17,184,25,192),(18,183,26,191),(19,182,27,190),(20,181,28,189),(21,180,29,188),(22,179,30,187),(23,178,31,186),(24,177,32,185),(33,122,41,114),(34,121,42,113),(35,120,43,128),(36,119,44,127),(37,118,45,126),(38,117,46,125),(39,116,47,124),(40,115,48,123),(49,306,57,314),(50,305,58,313),(51,320,59,312),(52,319,60,311),(53,318,61,310),(54,317,62,309),(55,316,63,308),(56,315,64,307),(65,433,73,441),(66,448,74,440),(67,447,75,439),(68,446,76,438),(69,445,77,437),(70,444,78,436),(71,443,79,435),(72,442,80,434),(81,458,89,450),(82,457,90,449),(83,456,91,464),(84,455,92,463),(85,454,93,462),(86,453,94,461),(87,452,95,460),(88,451,96,459),(97,254,105,246),(98,253,106,245),(99,252,107,244),(100,251,108,243),(101,250,109,242),(102,249,110,241),(103,248,111,256),(104,247,112,255),(129,173,137,165),(130,172,138,164),(131,171,139,163),(132,170,140,162),(133,169,141,161),(134,168,142,176),(135,167,143,175),(136,166,144,174),(145,223,153,215),(146,222,154,214),(147,221,155,213),(148,220,156,212),(149,219,157,211),(150,218,158,210),(151,217,159,209),(152,216,160,224),(193,402,201,410),(194,401,202,409),(195,416,203,408),(196,415,204,407),(197,414,205,406),(198,413,206,405),(199,412,207,404),(200,411,208,403),(225,264,233,272),(226,263,234,271),(227,262,235,270),(228,261,236,269),(229,260,237,268),(230,259,238,267),(231,258,239,266),(232,257,240,265),(289,330,297,322),(290,329,298,321),(291,328,299,336),(292,327,300,335),(293,326,301,334),(294,325,302,333),(295,324,303,332),(296,323,304,331),(337,365,345,357),(338,364,346,356),(339,363,347,355),(340,362,348,354),(341,361,349,353),(342,360,350,368),(343,359,351,367),(344,358,352,366),(369,480,377,472),(370,479,378,471),(371,478,379,470),(372,477,380,469),(373,476,381,468),(374,475,382,467),(375,474,383,466),(376,473,384,465),(385,428,393,420),(386,427,394,419),(387,426,395,418),(388,425,396,417),(389,424,397,432),(390,423,398,431),(391,422,399,430),(392,421,400,429)]])
90 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 20K | 20L | 24A | 24B | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 60A | 60B | 60C | 60D | 60E | ··· | 60L | 80A | ··· | 80P | 120A | ··· | 120H |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 2 | 8 | 24 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 24 | 24 | 24 | 24 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 6 | ··· | 6 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | D6 | D8 | C3⋊D4 | C5×S3 | Q32 | C5×D4 | S3×C10 | C5×D8 | C5×C3⋊D4 | C5×Q32 | D4⋊S3 | C3⋊Q32 | C5×D4⋊S3 | C5×C3⋊Q32 |
| kernel | C5×C3⋊Q32 | C5×C3⋊C16 | C5×Dic12 | C15×Q16 | C3⋊Q32 | C3⋊C16 | Dic12 | C3×Q16 | C5×Q16 | C60 | C40 | C30 | C20 | Q16 | C15 | C12 | C8 | C6 | C4 | C3 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C5×C3⋊Q32 ►in GL6(𝔽241)
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 11 | 230 | 0 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 54 | 0 | 0 |
| 0 | 0 | 214 | 58 | 0 | 0 |
| 0 | 0 | 0 | 0 | 71 | 89 |
| 0 | 0 | 0 | 0 | 160 | 170 |
| 138 | 113 | 0 | 0 | 0 | 0 |
| 113 | 103 | 0 | 0 | 0 | 0 |
| 0 | 0 | 204 | 165 | 0 | 0 |
| 0 | 0 | 240 | 37 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [98,0,0,0,0,0,0,98,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[11,11,0,0,0,0,230,11,0,0,0,0,0,0,112,214,0,0,0,0,54,58,0,0,0,0,0,0,71,160,0,0,0,0,89,170],[138,113,0,0,0,0,113,103,0,0,0,0,0,0,204,240,0,0,0,0,165,37,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;
C5×C3⋊Q32 in GAP, Magma, Sage, TeX
C_5\times C_3\rtimes Q_{32} % in TeX
G:=Group("C5xC3:Q32"); // GroupNames label
G:=SmallGroup(480,148);
// by ID
G=gap.SmallGroup(480,148);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,560,309,568,1683,850,192,4204,2111,102,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^3=c^16=1,d^2=c^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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