direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C5⋊Q32, C15⋊8Q32, C30.49D8, C24.51D10, C60.116D4, Dic20.2C6, C120.44C22, C5⋊2(C3×Q32), C8.7(C6×D5), Q16.(C3×D5), C40.5(C2×C6), C20.6(C3×D4), C5⋊2C16.1C6, C10.11(C3×D8), (C5×Q16).1C6, (C3×Q16).2D5, C6.27(D4⋊D5), (C15×Q16).2C2, C12.72(C5⋊D4), (C3×Dic20).4C2, C2.7(C3×D4⋊D5), C4.4(C3×C5⋊D4), (C3×C5⋊2C16).2C2, SmallGroup(480,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C5⋊Q32
G = < a,b,c,d | a3=b5=c16=1, d2=c8, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Smallest permutation representation of C3×C5⋊Q32
►Regular action on 480 points
Generators in S480
(1 153 86)(2 154 87)(3 155 88)(4 156 89)(5 157 90)(6 158 91)(7 159 92)(8 160 93)(9 145 94)(10 146 95)(11 147 96)(12 148 81)(13 149 82)(14 150 83)(15 151 84)(16 152 85)(17 476 286)(18 477 287)(19 478 288)(20 479 273)(21 480 274)(22 465 275)(23 466 276)(24 467 277)(25 468 278)(26 469 279)(27 470 280)(28 471 281)(29 472 282)(30 473 283)(31 474 284)(32 475 285)(33 415 396)(34 416 397)(35 401 398)(36 402 399)(37 403 400)(38 404 385)(39 405 386)(40 406 387)(41 407 388)(42 408 389)(43 409 390)(44 410 391)(45 411 392)(46 412 393)(47 413 394)(48 414 395)(49 321 452)(50 322 453)(51 323 454)(52 324 455)(53 325 456)(54 326 457)(55 327 458)(56 328 459)(57 329 460)(58 330 461)(59 331 462)(60 332 463)(61 333 464)(62 334 449)(63 335 450)(64 336 451)(65 293 220)(66 294 221)(67 295 222)(68 296 223)(69 297 224)(70 298 209)(71 299 210)(72 300 211)(73 301 212)(74 302 213)(75 303 214)(76 304 215)(77 289 216)(78 290 217)(79 291 218)(80 292 219)(97 137 364)(98 138 365)(99 139 366)(100 140 367)(101 141 368)(102 142 353)(103 143 354)(104 144 355)(105 129 356)(106 130 357)(107 131 358)(108 132 359)(109 133 360)(110 134 361)(111 135 362)(112 136 363)(113 269 241)(114 270 242)(115 271 243)(116 272 244)(117 257 245)(118 258 246)(119 259 247)(120 260 248)(121 261 249)(122 262 250)(123 263 251)(124 264 252)(125 265 253)(126 266 254)(127 267 255)(128 268 256)(161 207 318)(162 208 319)(163 193 320)(164 194 305)(165 195 306)(166 196 307)(167 197 308)(168 198 309)(169 199 310)(170 200 311)(171 201 312)(172 202 313)(173 203 314)(174 204 315)(175 205 316)(176 206 317)(177 418 229)(178 419 230)(179 420 231)(180 421 232)(181 422 233)(182 423 234)(183 424 235)(184 425 236)(185 426 237)(186 427 238)(187 428 239)(188 429 240)(189 430 225)(190 431 226)(191 432 227)(192 417 228)(337 376 443)(338 377 444)(339 378 445)(340 379 446)(341 380 447)(342 381 448)(343 382 433)(344 383 434)(345 384 435)(346 369 436)(347 370 437)(348 371 438)(349 372 439)(350 373 440)(351 374 441)(352 375 442)
(1 395 165 260 110)(2 111 261 166 396)(3 397 167 262 112)(4 97 263 168 398)(5 399 169 264 98)(6 99 265 170 400)(7 385 171 266 100)(8 101 267 172 386)(9 387 173 268 102)(10 103 269 174 388)(11 389 175 270 104)(12 105 271 176 390)(13 391 161 272 106)(14 107 257 162 392)(15 393 163 258 108)(16 109 259 164 394)(17 228 209 58 337)(18 338 59 210 229)(19 230 211 60 339)(20 340 61 212 231)(21 232 213 62 341)(22 342 63 214 233)(23 234 215 64 343)(24 344 49 216 235)(25 236 217 50 345)(26 346 51 218 237)(27 238 219 52 347)(28 348 53 220 239)(29 240 221 54 349)(30 350 55 222 225)(31 226 223 56 351)(32 352 57 224 227)(33 154 135 249 196)(34 197 250 136 155)(35 156 137 251 198)(36 199 252 138 157)(37 158 139 253 200)(38 201 254 140 159)(39 160 141 255 202)(40 203 256 142 145)(41 146 143 241 204)(42 205 242 144 147)(43 148 129 243 206)(44 207 244 130 149)(45 150 131 245 208)(46 193 246 132 151)(47 152 133 247 194)(48 195 248 134 153)(65 187 471 371 325)(66 326 372 472 188)(67 189 473 373 327)(68 328 374 474 190)(69 191 475 375 329)(70 330 376 476 192)(71 177 477 377 331)(72 332 378 478 178)(73 179 479 379 333)(74 334 380 480 180)(75 181 465 381 335)(76 336 382 466 182)(77 183 467 383 321)(78 322 384 468 184)(79 185 469 369 323)(80 324 370 470 186)(81 356 115 317 409)(82 410 318 116 357)(83 358 117 319 411)(84 412 320 118 359)(85 360 119 305 413)(86 414 306 120 361)(87 362 121 307 415)(88 416 308 122 363)(89 364 123 309 401)(90 402 310 124 365)(91 366 125 311 403)(92 404 312 126 367)(93 368 127 313 405)(94 406 314 128 353)(95 354 113 315 407)(96 408 316 114 355)(273 446 464 301 420)(274 421 302 449 447)(275 448 450 303 422)(276 423 304 451 433)(277 434 452 289 424)(278 425 290 453 435)(279 436 454 291 426)(280 427 292 455 437)(281 438 456 293 428)(282 429 294 457 439)(283 440 458 295 430)(284 431 296 459 441)(285 442 460 297 432)(286 417 298 461 443)(287 444 462 299 418)(288 419 300 463 445)
(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)
(1 289 9 297)(2 304 10 296)(3 303 11 295)(4 302 12 294)(5 301 13 293)(6 300 14 292)(7 299 15 291)(8 298 16 290)(17 194 25 202)(18 193 26 201)(19 208 27 200)(20 207 28 199)(21 206 29 198)(22 205 30 197)(23 204 31 196)(24 203 32 195)(33 234 41 226)(34 233 42 225)(35 232 43 240)(36 231 44 239)(37 230 45 238)(38 229 46 237)(39 228 47 236)(40 227 48 235)(49 142 57 134)(50 141 58 133)(51 140 59 132)(52 139 60 131)(53 138 61 130)(54 137 62 129)(55 136 63 144)(56 135 64 143)(65 90 73 82)(66 89 74 81)(67 88 75 96)(68 87 76 95)(69 86 77 94)(70 85 78 93)(71 84 79 92)(72 83 80 91)(97 449 105 457)(98 464 106 456)(99 463 107 455)(100 462 108 454)(101 461 109 453)(102 460 110 452)(103 459 111 451)(104 458 112 450)(113 374 121 382)(114 373 122 381)(115 372 123 380)(116 371 124 379)(117 370 125 378)(118 369 126 377)(119 384 127 376)(120 383 128 375)(145 224 153 216)(146 223 154 215)(147 222 155 214)(148 221 156 213)(149 220 157 212)(150 219 158 211)(151 218 159 210)(152 217 160 209)(161 281 169 273)(162 280 170 288)(163 279 171 287)(164 278 172 286)(165 277 173 285)(166 276 174 284)(167 275 175 283)(168 274 176 282)(177 412 185 404)(178 411 186 403)(179 410 187 402)(180 409 188 401)(181 408 189 416)(182 407 190 415)(183 406 191 414)(184 405 192 413)(241 351 249 343)(242 350 250 342)(243 349 251 341)(244 348 252 340)(245 347 253 339)(246 346 254 338)(247 345 255 337)(248 344 256 352)(257 437 265 445)(258 436 266 444)(259 435 267 443)(260 434 268 442)(261 433 269 441)(262 448 270 440)(263 447 271 439)(264 446 272 438)(305 468 313 476)(306 467 314 475)(307 466 315 474)(308 465 316 473)(309 480 317 472)(310 479 318 471)(311 478 319 470)(312 477 320 469)(321 353 329 361)(322 368 330 360)(323 367 331 359)(324 366 332 358)(325 365 333 357)(326 364 334 356)(327 363 335 355)(328 362 336 354)(385 418 393 426)(386 417 394 425)(387 432 395 424)(388 431 396 423)(389 430 397 422)(390 429 398 421)(391 428 399 420)(392 427 400 419)
G:=sub<Sym(480)| (1,153,86)(2,154,87)(3,155,88)(4,156,89)(5,157,90)(6,158,91)(7,159,92)(8,160,93)(9,145,94)(10,146,95)(11,147,96)(12,148,81)(13,149,82)(14,150,83)(15,151,84)(16,152,85)(17,476,286)(18,477,287)(19,478,288)(20,479,273)(21,480,274)(22,465,275)(23,466,276)(24,467,277)(25,468,278)(26,469,279)(27,470,280)(28,471,281)(29,472,282)(30,473,283)(31,474,284)(32,475,285)(33,415,396)(34,416,397)(35,401,398)(36,402,399)(37,403,400)(38,404,385)(39,405,386)(40,406,387)(41,407,388)(42,408,389)(43,409,390)(44,410,391)(45,411,392)(46,412,393)(47,413,394)(48,414,395)(49,321,452)(50,322,453)(51,323,454)(52,324,455)(53,325,456)(54,326,457)(55,327,458)(56,328,459)(57,329,460)(58,330,461)(59,331,462)(60,332,463)(61,333,464)(62,334,449)(63,335,450)(64,336,451)(65,293,220)(66,294,221)(67,295,222)(68,296,223)(69,297,224)(70,298,209)(71,299,210)(72,300,211)(73,301,212)(74,302,213)(75,303,214)(76,304,215)(77,289,216)(78,290,217)(79,291,218)(80,292,219)(97,137,364)(98,138,365)(99,139,366)(100,140,367)(101,141,368)(102,142,353)(103,143,354)(104,144,355)(105,129,356)(106,130,357)(107,131,358)(108,132,359)(109,133,360)(110,134,361)(111,135,362)(112,136,363)(113,269,241)(114,270,242)(115,271,243)(116,272,244)(117,257,245)(118,258,246)(119,259,247)(120,260,248)(121,261,249)(122,262,250)(123,263,251)(124,264,252)(125,265,253)(126,266,254)(127,267,255)(128,268,256)(161,207,318)(162,208,319)(163,193,320)(164,194,305)(165,195,306)(166,196,307)(167,197,308)(168,198,309)(169,199,310)(170,200,311)(171,201,312)(172,202,313)(173,203,314)(174,204,315)(175,205,316)(176,206,317)(177,418,229)(178,419,230)(179,420,231)(180,421,232)(181,422,233)(182,423,234)(183,424,235)(184,425,236)(185,426,237)(186,427,238)(187,428,239)(188,429,240)(189,430,225)(190,431,226)(191,432,227)(192,417,228)(337,376,443)(338,377,444)(339,378,445)(340,379,446)(341,380,447)(342,381,448)(343,382,433)(344,383,434)(345,384,435)(346,369,436)(347,370,437)(348,371,438)(349,372,439)(350,373,440)(351,374,441)(352,375,442), (1,395,165,260,110)(2,111,261,166,396)(3,397,167,262,112)(4,97,263,168,398)(5,399,169,264,98)(6,99,265,170,400)(7,385,171,266,100)(8,101,267,172,386)(9,387,173,268,102)(10,103,269,174,388)(11,389,175,270,104)(12,105,271,176,390)(13,391,161,272,106)(14,107,257,162,392)(15,393,163,258,108)(16,109,259,164,394)(17,228,209,58,337)(18,338,59,210,229)(19,230,211,60,339)(20,340,61,212,231)(21,232,213,62,341)(22,342,63,214,233)(23,234,215,64,343)(24,344,49,216,235)(25,236,217,50,345)(26,346,51,218,237)(27,238,219,52,347)(28,348,53,220,239)(29,240,221,54,349)(30,350,55,222,225)(31,226,223,56,351)(32,352,57,224,227)(33,154,135,249,196)(34,197,250,136,155)(35,156,137,251,198)(36,199,252,138,157)(37,158,139,253,200)(38,201,254,140,159)(39,160,141,255,202)(40,203,256,142,145)(41,146,143,241,204)(42,205,242,144,147)(43,148,129,243,206)(44,207,244,130,149)(45,150,131,245,208)(46,193,246,132,151)(47,152,133,247,194)(48,195,248,134,153)(65,187,471,371,325)(66,326,372,472,188)(67,189,473,373,327)(68,328,374,474,190)(69,191,475,375,329)(70,330,376,476,192)(71,177,477,377,331)(72,332,378,478,178)(73,179,479,379,333)(74,334,380,480,180)(75,181,465,381,335)(76,336,382,466,182)(77,183,467,383,321)(78,322,384,468,184)(79,185,469,369,323)(80,324,370,470,186)(81,356,115,317,409)(82,410,318,116,357)(83,358,117,319,411)(84,412,320,118,359)(85,360,119,305,413)(86,414,306,120,361)(87,362,121,307,415)(88,416,308,122,363)(89,364,123,309,401)(90,402,310,124,365)(91,366,125,311,403)(92,404,312,126,367)(93,368,127,313,405)(94,406,314,128,353)(95,354,113,315,407)(96,408,316,114,355)(273,446,464,301,420)(274,421,302,449,447)(275,448,450,303,422)(276,423,304,451,433)(277,434,452,289,424)(278,425,290,453,435)(279,436,454,291,426)(280,427,292,455,437)(281,438,456,293,428)(282,429,294,457,439)(283,440,458,295,430)(284,431,296,459,441)(285,442,460,297,432)(286,417,298,461,443)(287,444,462,299,418)(288,419,300,463,445), (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), (1,289,9,297)(2,304,10,296)(3,303,11,295)(4,302,12,294)(5,301,13,293)(6,300,14,292)(7,299,15,291)(8,298,16,290)(17,194,25,202)(18,193,26,201)(19,208,27,200)(20,207,28,199)(21,206,29,198)(22,205,30,197)(23,204,31,196)(24,203,32,195)(33,234,41,226)(34,233,42,225)(35,232,43,240)(36,231,44,239)(37,230,45,238)(38,229,46,237)(39,228,47,236)(40,227,48,235)(49,142,57,134)(50,141,58,133)(51,140,59,132)(52,139,60,131)(53,138,61,130)(54,137,62,129)(55,136,63,144)(56,135,64,143)(65,90,73,82)(66,89,74,81)(67,88,75,96)(68,87,76,95)(69,86,77,94)(70,85,78,93)(71,84,79,92)(72,83,80,91)(97,449,105,457)(98,464,106,456)(99,463,107,455)(100,462,108,454)(101,461,109,453)(102,460,110,452)(103,459,111,451)(104,458,112,450)(113,374,121,382)(114,373,122,381)(115,372,123,380)(116,371,124,379)(117,370,125,378)(118,369,126,377)(119,384,127,376)(120,383,128,375)(145,224,153,216)(146,223,154,215)(147,222,155,214)(148,221,156,213)(149,220,157,212)(150,219,158,211)(151,218,159,210)(152,217,160,209)(161,281,169,273)(162,280,170,288)(163,279,171,287)(164,278,172,286)(165,277,173,285)(166,276,174,284)(167,275,175,283)(168,274,176,282)(177,412,185,404)(178,411,186,403)(179,410,187,402)(180,409,188,401)(181,408,189,416)(182,407,190,415)(183,406,191,414)(184,405,192,413)(241,351,249,343)(242,350,250,342)(243,349,251,341)(244,348,252,340)(245,347,253,339)(246,346,254,338)(247,345,255,337)(248,344,256,352)(257,437,265,445)(258,436,266,444)(259,435,267,443)(260,434,268,442)(261,433,269,441)(262,448,270,440)(263,447,271,439)(264,446,272,438)(305,468,313,476)(306,467,314,475)(307,466,315,474)(308,465,316,473)(309,480,317,472)(310,479,318,471)(311,478,319,470)(312,477,320,469)(321,353,329,361)(322,368,330,360)(323,367,331,359)(324,366,332,358)(325,365,333,357)(326,364,334,356)(327,363,335,355)(328,362,336,354)(385,418,393,426)(386,417,394,425)(387,432,395,424)(388,431,396,423)(389,430,397,422)(390,429,398,421)(391,428,399,420)(392,427,400,419)>;
G:=Group( (1,153,86)(2,154,87)(3,155,88)(4,156,89)(5,157,90)(6,158,91)(7,159,92)(8,160,93)(9,145,94)(10,146,95)(11,147,96)(12,148,81)(13,149,82)(14,150,83)(15,151,84)(16,152,85)(17,476,286)(18,477,287)(19,478,288)(20,479,273)(21,480,274)(22,465,275)(23,466,276)(24,467,277)(25,468,278)(26,469,279)(27,470,280)(28,471,281)(29,472,282)(30,473,283)(31,474,284)(32,475,285)(33,415,396)(34,416,397)(35,401,398)(36,402,399)(37,403,400)(38,404,385)(39,405,386)(40,406,387)(41,407,388)(42,408,389)(43,409,390)(44,410,391)(45,411,392)(46,412,393)(47,413,394)(48,414,395)(49,321,452)(50,322,453)(51,323,454)(52,324,455)(53,325,456)(54,326,457)(55,327,458)(56,328,459)(57,329,460)(58,330,461)(59,331,462)(60,332,463)(61,333,464)(62,334,449)(63,335,450)(64,336,451)(65,293,220)(66,294,221)(67,295,222)(68,296,223)(69,297,224)(70,298,209)(71,299,210)(72,300,211)(73,301,212)(74,302,213)(75,303,214)(76,304,215)(77,289,216)(78,290,217)(79,291,218)(80,292,219)(97,137,364)(98,138,365)(99,139,366)(100,140,367)(101,141,368)(102,142,353)(103,143,354)(104,144,355)(105,129,356)(106,130,357)(107,131,358)(108,132,359)(109,133,360)(110,134,361)(111,135,362)(112,136,363)(113,269,241)(114,270,242)(115,271,243)(116,272,244)(117,257,245)(118,258,246)(119,259,247)(120,260,248)(121,261,249)(122,262,250)(123,263,251)(124,264,252)(125,265,253)(126,266,254)(127,267,255)(128,268,256)(161,207,318)(162,208,319)(163,193,320)(164,194,305)(165,195,306)(166,196,307)(167,197,308)(168,198,309)(169,199,310)(170,200,311)(171,201,312)(172,202,313)(173,203,314)(174,204,315)(175,205,316)(176,206,317)(177,418,229)(178,419,230)(179,420,231)(180,421,232)(181,422,233)(182,423,234)(183,424,235)(184,425,236)(185,426,237)(186,427,238)(187,428,239)(188,429,240)(189,430,225)(190,431,226)(191,432,227)(192,417,228)(337,376,443)(338,377,444)(339,378,445)(340,379,446)(341,380,447)(342,381,448)(343,382,433)(344,383,434)(345,384,435)(346,369,436)(347,370,437)(348,371,438)(349,372,439)(350,373,440)(351,374,441)(352,375,442), (1,395,165,260,110)(2,111,261,166,396)(3,397,167,262,112)(4,97,263,168,398)(5,399,169,264,98)(6,99,265,170,400)(7,385,171,266,100)(8,101,267,172,386)(9,387,173,268,102)(10,103,269,174,388)(11,389,175,270,104)(12,105,271,176,390)(13,391,161,272,106)(14,107,257,162,392)(15,393,163,258,108)(16,109,259,164,394)(17,228,209,58,337)(18,338,59,210,229)(19,230,211,60,339)(20,340,61,212,231)(21,232,213,62,341)(22,342,63,214,233)(23,234,215,64,343)(24,344,49,216,235)(25,236,217,50,345)(26,346,51,218,237)(27,238,219,52,347)(28,348,53,220,239)(29,240,221,54,349)(30,350,55,222,225)(31,226,223,56,351)(32,352,57,224,227)(33,154,135,249,196)(34,197,250,136,155)(35,156,137,251,198)(36,199,252,138,157)(37,158,139,253,200)(38,201,254,140,159)(39,160,141,255,202)(40,203,256,142,145)(41,146,143,241,204)(42,205,242,144,147)(43,148,129,243,206)(44,207,244,130,149)(45,150,131,245,208)(46,193,246,132,151)(47,152,133,247,194)(48,195,248,134,153)(65,187,471,371,325)(66,326,372,472,188)(67,189,473,373,327)(68,328,374,474,190)(69,191,475,375,329)(70,330,376,476,192)(71,177,477,377,331)(72,332,378,478,178)(73,179,479,379,333)(74,334,380,480,180)(75,181,465,381,335)(76,336,382,466,182)(77,183,467,383,321)(78,322,384,468,184)(79,185,469,369,323)(80,324,370,470,186)(81,356,115,317,409)(82,410,318,116,357)(83,358,117,319,411)(84,412,320,118,359)(85,360,119,305,413)(86,414,306,120,361)(87,362,121,307,415)(88,416,308,122,363)(89,364,123,309,401)(90,402,310,124,365)(91,366,125,311,403)(92,404,312,126,367)(93,368,127,313,405)(94,406,314,128,353)(95,354,113,315,407)(96,408,316,114,355)(273,446,464,301,420)(274,421,302,449,447)(275,448,450,303,422)(276,423,304,451,433)(277,434,452,289,424)(278,425,290,453,435)(279,436,454,291,426)(280,427,292,455,437)(281,438,456,293,428)(282,429,294,457,439)(283,440,458,295,430)(284,431,296,459,441)(285,442,460,297,432)(286,417,298,461,443)(287,444,462,299,418)(288,419,300,463,445), (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), (1,289,9,297)(2,304,10,296)(3,303,11,295)(4,302,12,294)(5,301,13,293)(6,300,14,292)(7,299,15,291)(8,298,16,290)(17,194,25,202)(18,193,26,201)(19,208,27,200)(20,207,28,199)(21,206,29,198)(22,205,30,197)(23,204,31,196)(24,203,32,195)(33,234,41,226)(34,233,42,225)(35,232,43,240)(36,231,44,239)(37,230,45,238)(38,229,46,237)(39,228,47,236)(40,227,48,235)(49,142,57,134)(50,141,58,133)(51,140,59,132)(52,139,60,131)(53,138,61,130)(54,137,62,129)(55,136,63,144)(56,135,64,143)(65,90,73,82)(66,89,74,81)(67,88,75,96)(68,87,76,95)(69,86,77,94)(70,85,78,93)(71,84,79,92)(72,83,80,91)(97,449,105,457)(98,464,106,456)(99,463,107,455)(100,462,108,454)(101,461,109,453)(102,460,110,452)(103,459,111,451)(104,458,112,450)(113,374,121,382)(114,373,122,381)(115,372,123,380)(116,371,124,379)(117,370,125,378)(118,369,126,377)(119,384,127,376)(120,383,128,375)(145,224,153,216)(146,223,154,215)(147,222,155,214)(148,221,156,213)(149,220,157,212)(150,219,158,211)(151,218,159,210)(152,217,160,209)(161,281,169,273)(162,280,170,288)(163,279,171,287)(164,278,172,286)(165,277,173,285)(166,276,174,284)(167,275,175,283)(168,274,176,282)(177,412,185,404)(178,411,186,403)(179,410,187,402)(180,409,188,401)(181,408,189,416)(182,407,190,415)(183,406,191,414)(184,405,192,413)(241,351,249,343)(242,350,250,342)(243,349,251,341)(244,348,252,340)(245,347,253,339)(246,346,254,338)(247,345,255,337)(248,344,256,352)(257,437,265,445)(258,436,266,444)(259,435,267,443)(260,434,268,442)(261,433,269,441)(262,448,270,440)(263,447,271,439)(264,446,272,438)(305,468,313,476)(306,467,314,475)(307,466,315,474)(308,465,316,473)(309,480,317,472)(310,479,318,471)(311,478,319,470)(312,477,320,469)(321,353,329,361)(322,368,330,360)(323,367,331,359)(324,366,332,358)(325,365,333,357)(326,364,334,356)(327,363,335,355)(328,362,336,354)(385,418,393,426)(386,417,394,425)(387,432,395,424)(388,431,396,423)(389,430,397,422)(390,429,398,421)(391,428,399,420)(392,427,400,419) );
G=PermutationGroup([[(1,153,86),(2,154,87),(3,155,88),(4,156,89),(5,157,90),(6,158,91),(7,159,92),(8,160,93),(9,145,94),(10,146,95),(11,147,96),(12,148,81),(13,149,82),(14,150,83),(15,151,84),(16,152,85),(17,476,286),(18,477,287),(19,478,288),(20,479,273),(21,480,274),(22,465,275),(23,466,276),(24,467,277),(25,468,278),(26,469,279),(27,470,280),(28,471,281),(29,472,282),(30,473,283),(31,474,284),(32,475,285),(33,415,396),(34,416,397),(35,401,398),(36,402,399),(37,403,400),(38,404,385),(39,405,386),(40,406,387),(41,407,388),(42,408,389),(43,409,390),(44,410,391),(45,411,392),(46,412,393),(47,413,394),(48,414,395),(49,321,452),(50,322,453),(51,323,454),(52,324,455),(53,325,456),(54,326,457),(55,327,458),(56,328,459),(57,329,460),(58,330,461),(59,331,462),(60,332,463),(61,333,464),(62,334,449),(63,335,450),(64,336,451),(65,293,220),(66,294,221),(67,295,222),(68,296,223),(69,297,224),(70,298,209),(71,299,210),(72,300,211),(73,301,212),(74,302,213),(75,303,214),(76,304,215),(77,289,216),(78,290,217),(79,291,218),(80,292,219),(97,137,364),(98,138,365),(99,139,366),(100,140,367),(101,141,368),(102,142,353),(103,143,354),(104,144,355),(105,129,356),(106,130,357),(107,131,358),(108,132,359),(109,133,360),(110,134,361),(111,135,362),(112,136,363),(113,269,241),(114,270,242),(115,271,243),(116,272,244),(117,257,245),(118,258,246),(119,259,247),(120,260,248),(121,261,249),(122,262,250),(123,263,251),(124,264,252),(125,265,253),(126,266,254),(127,267,255),(128,268,256),(161,207,318),(162,208,319),(163,193,320),(164,194,305),(165,195,306),(166,196,307),(167,197,308),(168,198,309),(169,199,310),(170,200,311),(171,201,312),(172,202,313),(173,203,314),(174,204,315),(175,205,316),(176,206,317),(177,418,229),(178,419,230),(179,420,231),(180,421,232),(181,422,233),(182,423,234),(183,424,235),(184,425,236),(185,426,237),(186,427,238),(187,428,239),(188,429,240),(189,430,225),(190,431,226),(191,432,227),(192,417,228),(337,376,443),(338,377,444),(339,378,445),(340,379,446),(341,380,447),(342,381,448),(343,382,433),(344,383,434),(345,384,435),(346,369,436),(347,370,437),(348,371,438),(349,372,439),(350,373,440),(351,374,441),(352,375,442)], [(1,395,165,260,110),(2,111,261,166,396),(3,397,167,262,112),(4,97,263,168,398),(5,399,169,264,98),(6,99,265,170,400),(7,385,171,266,100),(8,101,267,172,386),(9,387,173,268,102),(10,103,269,174,388),(11,389,175,270,104),(12,105,271,176,390),(13,391,161,272,106),(14,107,257,162,392),(15,393,163,258,108),(16,109,259,164,394),(17,228,209,58,337),(18,338,59,210,229),(19,230,211,60,339),(20,340,61,212,231),(21,232,213,62,341),(22,342,63,214,233),(23,234,215,64,343),(24,344,49,216,235),(25,236,217,50,345),(26,346,51,218,237),(27,238,219,52,347),(28,348,53,220,239),(29,240,221,54,349),(30,350,55,222,225),(31,226,223,56,351),(32,352,57,224,227),(33,154,135,249,196),(34,197,250,136,155),(35,156,137,251,198),(36,199,252,138,157),(37,158,139,253,200),(38,201,254,140,159),(39,160,141,255,202),(40,203,256,142,145),(41,146,143,241,204),(42,205,242,144,147),(43,148,129,243,206),(44,207,244,130,149),(45,150,131,245,208),(46,193,246,132,151),(47,152,133,247,194),(48,195,248,134,153),(65,187,471,371,325),(66,326,372,472,188),(67,189,473,373,327),(68,328,374,474,190),(69,191,475,375,329),(70,330,376,476,192),(71,177,477,377,331),(72,332,378,478,178),(73,179,479,379,333),(74,334,380,480,180),(75,181,465,381,335),(76,336,382,466,182),(77,183,467,383,321),(78,322,384,468,184),(79,185,469,369,323),(80,324,370,470,186),(81,356,115,317,409),(82,410,318,116,357),(83,358,117,319,411),(84,412,320,118,359),(85,360,119,305,413),(86,414,306,120,361),(87,362,121,307,415),(88,416,308,122,363),(89,364,123,309,401),(90,402,310,124,365),(91,366,125,311,403),(92,404,312,126,367),(93,368,127,313,405),(94,406,314,128,353),(95,354,113,315,407),(96,408,316,114,355),(273,446,464,301,420),(274,421,302,449,447),(275,448,450,303,422),(276,423,304,451,433),(277,434,452,289,424),(278,425,290,453,435),(279,436,454,291,426),(280,427,292,455,437),(281,438,456,293,428),(282,429,294,457,439),(283,440,458,295,430),(284,431,296,459,441),(285,442,460,297,432),(286,417,298,461,443),(287,444,462,299,418),(288,419,300,463,445)], [(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)], [(1,289,9,297),(2,304,10,296),(3,303,11,295),(4,302,12,294),(5,301,13,293),(6,300,14,292),(7,299,15,291),(8,298,16,290),(17,194,25,202),(18,193,26,201),(19,208,27,200),(20,207,28,199),(21,206,29,198),(22,205,30,197),(23,204,31,196),(24,203,32,195),(33,234,41,226),(34,233,42,225),(35,232,43,240),(36,231,44,239),(37,230,45,238),(38,229,46,237),(39,228,47,236),(40,227,48,235),(49,142,57,134),(50,141,58,133),(51,140,59,132),(52,139,60,131),(53,138,61,130),(54,137,62,129),(55,136,63,144),(56,135,64,143),(65,90,73,82),(66,89,74,81),(67,88,75,96),(68,87,76,95),(69,86,77,94),(70,85,78,93),(71,84,79,92),(72,83,80,91),(97,449,105,457),(98,464,106,456),(99,463,107,455),(100,462,108,454),(101,461,109,453),(102,460,110,452),(103,459,111,451),(104,458,112,450),(113,374,121,382),(114,373,122,381),(115,372,123,380),(116,371,124,379),(117,370,125,378),(118,369,126,377),(119,384,127,376),(120,383,128,375),(145,224,153,216),(146,223,154,215),(147,222,155,214),(148,221,156,213),(149,220,157,212),(150,219,158,211),(151,218,159,210),(152,217,160,209),(161,281,169,273),(162,280,170,288),(163,279,171,287),(164,278,172,286),(165,277,173,285),(166,276,174,284),(167,275,175,283),(168,274,176,282),(177,412,185,404),(178,411,186,403),(179,410,187,402),(180,409,188,401),(181,408,189,416),(182,407,190,415),(183,406,191,414),(184,405,192,413),(241,351,249,343),(242,350,250,342),(243,349,251,341),(244,348,252,340),(245,347,253,339),(246,346,254,338),(247,345,255,337),(248,344,256,352),(257,437,265,445),(258,436,266,444),(259,435,267,443),(260,434,268,442),(261,433,269,441),(262,448,270,440),(263,447,271,439),(264,446,272,438),(305,468,313,476),(306,467,314,475),(307,466,315,474),(308,465,316,473),(309,480,317,472),(310,479,318,471),(311,478,319,470),(312,477,320,469),(321,353,329,361),(322,368,330,360),(323,367,331,359),(324,366,332,358),(325,365,333,357),(326,364,334,356),(327,363,335,355),(328,362,336,354),(385,418,393,426),(386,417,394,425),(387,432,395,424),(388,431,396,423),(389,430,397,422),(390,429,398,421),(391,428,399,420),(392,427,400,419)]])
75 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | 40B | 40C | 40D | 48A | ··· | 48H | 60A | 60B | 60C | 60D | 60E | ··· | 60L | 120A | ··· | 120H |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 48 | ··· | 48 | 60 | 60 | 60 | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 2 | 8 | 40 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 40 | 40 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
75 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 | C3 | C6 | C6 | C6 | D4 | D5 | D8 | D10 | C3×D4 | C3×D5 | Q32 | C5⋊D4 | C3×D8 | C6×D5 | C3×Q32 | C3×C5⋊D4 | D4⋊D5 | C5⋊Q32 | C3×D4⋊D5 | C3×C5⋊Q32 |
| kernel | C3×C5⋊Q32 | C3×C5⋊2C16 | C3×Dic20 | C15×Q16 | C5⋊Q32 | C5⋊2C16 | Dic20 | C5×Q16 | C60 | C3×Q16 | C30 | C24 | C20 | Q16 | C15 | C12 | C10 | C8 | C5 | C4 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 4 | 4 | 8 |
Matrix representation of C3×C5⋊Q32 ►in GL4(𝔽241) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 240 | 0 | 0 |
| 1 | 189 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 132 | 130 | 0 | 0 |
| 5 | 109 | 0 | 0 |
| 0 | 0 | 214 | 156 |
| 0 | 0 | 85 | 214 |
| 214 | 103 | 0 | 0 |
| 138 | 27 | 0 | 0 |
| 0 | 0 | 238 | 183 |
| 0 | 0 | 183 | 3 |
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,240,189,0,0,0,0,1,0,0,0,0,1],[132,5,0,0,130,109,0,0,0,0,214,85,0,0,156,214],[214,138,0,0,103,27,0,0,0,0,238,183,0,0,183,3] >;
C3×C5⋊Q32 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes Q_{32} % in TeX
G:=Group("C3xC5:Q32"); // GroupNames label
G:=SmallGroup(480,107);
// by ID
G=gap.SmallGroup(480,107);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,336,197,344,1011,514,192,2524,1271,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^5=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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