direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C4⋊Q8, C60⋊14Q8, C60.197D4, C42.5C30, C4⋊(Q8×C15), C20⋊4(C3×Q8), C12⋊4(C5×Q8), C4⋊C4.5C30, C4.5(D4×C15), C2.5(Q8×C30), (C4×C20).11C6, (C4×C60).23C2, C2.10(D4×C30), C10.73(C6×D4), C20.40(C3×D4), C12.40(C5×D4), C6.73(D4×C10), (C6×Q8).8C10, (C2×Q8).5C30, C10.22(C6×Q8), C6.22(Q8×C10), (C4×C12).11C10, C30.456(C2×D4), (Q8×C30).18C2, (Q8×C10).12C6, C30.120(C2×Q8), (C2×C60).439C22, (C2×C30).463C23, C22.18(C22×C30), (C5×C4⋊C4).12C6, (C2×C4).6(C2×C30), (C3×C4⋊C4).12C10, (C15×C4⋊C4).26C2, (C2×C20).68(C2×C6), (C2×C12).67(C2×C10), (C2×C10).83(C22×C6), (C2×C6).83(C22×C10), SmallGroup(480,933)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C4⋊Q8
G = < a,b,c,d | a15=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 168 in 136 conjugacy classes, 104 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C10, C10, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, C20, C20, C2×C10, C2×C12, C2×C12, C3×Q8, C30, C30, C4⋊Q8, C2×C20, C2×C20, C5×Q8, C4×C12, C3×C4⋊C4, C6×Q8, C60, C60, C2×C30, C4×C20, C5×C4⋊C4, Q8×C10, C3×C4⋊Q8, C2×C60, C2×C60, Q8×C15, C5×C4⋊Q8, C4×C60, C15×C4⋊C4, Q8×C30, C15×C4⋊Q8
Quotients: C1, C2, C3, C22, C5, C6, D4, Q8, C23, C10, C2×C6, C15, C2×D4, C2×Q8, C2×C10, C3×D4, C3×Q8, C22×C6, C30, C4⋊Q8, C5×D4, C5×Q8, C22×C10, C6×D4, C6×Q8, C2×C30, D4×C10, Q8×C10, C3×C4⋊Q8, D4×C15, Q8×C15, C22×C30, C5×C4⋊Q8, D4×C30, Q8×C30, C15×C4⋊Q8
►Regular action on 480 points
Generators in S480
(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)
(1 309 295 354)(2 310 296 355)(3 311 297 356)(4 312 298 357)(5 313 299 358)(6 314 300 359)(7 315 286 360)(8 301 287 346)(9 302 288 347)(10 303 289 348)(11 304 290 349)(12 305 291 350)(13 306 292 351)(14 307 293 352)(15 308 294 353)(16 157 48 411)(17 158 49 412)(18 159 50 413)(19 160 51 414)(20 161 52 415)(21 162 53 416)(22 163 54 417)(23 164 55 418)(24 165 56 419)(25 151 57 420)(26 152 58 406)(27 153 59 407)(28 154 60 408)(29 155 46 409)(30 156 47 410)(31 95 185 335)(32 96 186 336)(33 97 187 337)(34 98 188 338)(35 99 189 339)(36 100 190 340)(37 101 191 341)(38 102 192 342)(39 103 193 343)(40 104 194 344)(41 105 195 345)(42 91 181 331)(43 92 182 332)(44 93 183 333)(45 94 184 334)(61 241 90 172)(62 242 76 173)(63 243 77 174)(64 244 78 175)(65 245 79 176)(66 246 80 177)(67 247 81 178)(68 248 82 179)(69 249 83 180)(70 250 84 166)(71 251 85 167)(72 252 86 168)(73 253 87 169)(74 254 88 170)(75 255 89 171)(106 427 475 363)(107 428 476 364)(108 429 477 365)(109 430 478 366)(110 431 479 367)(111 432 480 368)(112 433 466 369)(113 434 467 370)(114 435 468 371)(115 421 469 372)(116 422 470 373)(117 423 471 374)(118 424 472 375)(119 425 473 361)(120 426 474 362)(121 217 258 231)(122 218 259 232)(123 219 260 233)(124 220 261 234)(125 221 262 235)(126 222 263 236)(127 223 264 237)(128 224 265 238)(129 225 266 239)(130 211 267 240)(131 212 268 226)(132 213 269 227)(133 214 270 228)(134 215 256 229)(135 216 257 230)(136 393 387 280)(137 394 388 281)(138 395 389 282)(139 396 390 283)(140 397 376 284)(141 398 377 285)(142 399 378 271)(143 400 379 272)(144 401 380 273)(145 402 381 274)(146 403 382 275)(147 404 383 276)(148 405 384 277)(149 391 385 278)(150 392 386 279)(196 318 448 462)(197 319 449 463)(198 320 450 464)(199 321 436 465)(200 322 437 451)(201 323 438 452)(202 324 439 453)(203 325 440 454)(204 326 441 455)(205 327 442 456)(206 328 443 457)(207 329 444 458)(208 330 445 459)(209 316 446 460)(210 317 447 461)
(1 437 138 268)(2 438 139 269)(3 439 140 270)(4 440 141 256)(5 441 142 257)(6 442 143 258)(7 443 144 259)(8 444 145 260)(9 445 146 261)(10 446 147 262)(11 447 148 263)(12 448 149 264)(13 449 150 265)(14 450 136 266)(15 436 137 267)(16 185 80 117)(17 186 81 118)(18 187 82 119)(19 188 83 120)(20 189 84 106)(21 190 85 107)(22 191 86 108)(23 192 87 109)(24 193 88 110)(25 194 89 111)(26 195 90 112)(27 181 76 113)(28 182 77 114)(29 183 78 115)(30 184 79 116)(31 66 471 48)(32 67 472 49)(33 68 473 50)(34 69 474 51)(35 70 475 52)(36 71 476 53)(37 72 477 54)(38 73 478 55)(39 74 479 56)(40 75 480 57)(41 61 466 58)(42 62 467 59)(43 63 468 60)(44 64 469 46)(45 65 470 47)(91 242 370 407)(92 243 371 408)(93 244 372 409)(94 245 373 410)(95 246 374 411)(96 247 375 412)(97 248 361 413)(98 249 362 414)(99 250 363 415)(100 251 364 416)(101 252 365 417)(102 253 366 418)(103 254 367 419)(104 255 368 420)(105 241 369 406)(121 300 205 379)(122 286 206 380)(123 287 207 381)(124 288 208 382)(125 289 209 383)(126 290 210 384)(127 291 196 385)(128 292 197 386)(129 293 198 387)(130 294 199 388)(131 295 200 389)(132 296 201 390)(133 297 202 376)(134 298 203 377)(135 299 204 378)(151 344 171 432)(152 345 172 433)(153 331 173 434)(154 332 174 435)(155 333 175 421)(156 334 176 422)(157 335 177 423)(158 336 178 424)(159 337 179 425)(160 338 180 426)(161 339 166 427)(162 340 167 428)(163 341 168 429)(164 342 169 430)(165 343 170 431)(211 353 321 281)(212 354 322 282)(213 355 323 283)(214 356 324 284)(215 357 325 285)(216 358 326 271)(217 359 327 272)(218 360 328 273)(219 346 329 274)(220 347 330 275)(221 348 316 276)(222 349 317 277)(223 350 318 278)(224 351 319 279)(225 352 320 280)(226 309 451 395)(227 310 452 396)(228 311 453 397)(229 312 454 398)(230 313 455 399)(231 314 456 400)(232 315 457 401)(233 301 458 402)(234 302 459 403)(235 303 460 404)(236 304 461 405)(237 305 462 391)(238 306 463 392)(239 307 464 393)(240 308 465 394)
(1 253 138 418)(2 254 139 419)(3 255 140 420)(4 241 141 406)(5 242 142 407)(6 243 143 408)(7 244 144 409)(8 245 145 410)(9 246 146 411)(10 247 147 412)(11 248 148 413)(12 249 149 414)(13 250 150 415)(14 251 136 416)(15 252 137 417)(16 347 80 275)(17 348 81 276)(18 349 82 277)(19 350 83 278)(20 351 84 279)(21 352 85 280)(22 353 86 281)(23 354 87 282)(24 355 88 283)(25 356 89 284)(26 357 90 285)(27 358 76 271)(28 359 77 272)(29 360 78 273)(30 346 79 274)(31 234 471 459)(32 235 472 460)(33 236 473 461)(34 237 474 462)(35 238 475 463)(36 239 476 464)(37 240 477 465)(38 226 478 451)(39 227 479 452)(40 228 480 453)(41 229 466 454)(42 230 467 455)(43 231 468 456)(44 232 469 457)(45 233 470 458)(46 315 64 401)(47 301 65 402)(48 302 66 403)(49 303 67 404)(50 304 68 405)(51 305 69 391)(52 306 70 392)(53 307 71 393)(54 308 72 394)(55 309 73 395)(56 310 74 396)(57 311 75 397)(58 312 61 398)(59 313 62 399)(60 314 63 400)(91 257 370 441)(92 258 371 442)(93 259 372 443)(94 260 373 444)(95 261 374 445)(96 262 375 446)(97 263 361 447)(98 264 362 448)(99 265 363 449)(100 266 364 450)(101 267 365 436)(102 268 366 437)(103 269 367 438)(104 270 368 439)(105 256 369 440)(106 319 189 224)(107 320 190 225)(108 321 191 211)(109 322 192 212)(110 323 193 213)(111 324 194 214)(112 325 195 215)(113 326 181 216)(114 327 182 217)(115 328 183 218)(116 329 184 219)(117 330 185 220)(118 316 186 221)(119 317 187 222)(120 318 188 223)(121 435 205 332)(122 421 206 333)(123 422 207 334)(124 423 208 335)(125 424 209 336)(126 425 210 337)(127 426 196 338)(128 427 197 339)(129 428 198 340)(130 429 199 341)(131 430 200 342)(132 431 201 343)(133 432 202 344)(134 433 203 345)(135 434 204 331)(151 297 171 376)(152 298 172 377)(153 299 173 378)(154 300 174 379)(155 286 175 380)(156 287 176 381)(157 288 177 382)(158 289 178 383)(159 290 179 384)(160 291 180 385)(161 292 166 386)(162 293 167 387)(163 294 168 388)(164 295 169 389)(165 296 170 390)
G:=sub<Sym(480)| (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), (1,309,295,354)(2,310,296,355)(3,311,297,356)(4,312,298,357)(5,313,299,358)(6,314,300,359)(7,315,286,360)(8,301,287,346)(9,302,288,347)(10,303,289,348)(11,304,290,349)(12,305,291,350)(13,306,292,351)(14,307,293,352)(15,308,294,353)(16,157,48,411)(17,158,49,412)(18,159,50,413)(19,160,51,414)(20,161,52,415)(21,162,53,416)(22,163,54,417)(23,164,55,418)(24,165,56,419)(25,151,57,420)(26,152,58,406)(27,153,59,407)(28,154,60,408)(29,155,46,409)(30,156,47,410)(31,95,185,335)(32,96,186,336)(33,97,187,337)(34,98,188,338)(35,99,189,339)(36,100,190,340)(37,101,191,341)(38,102,192,342)(39,103,193,343)(40,104,194,344)(41,105,195,345)(42,91,181,331)(43,92,182,332)(44,93,183,333)(45,94,184,334)(61,241,90,172)(62,242,76,173)(63,243,77,174)(64,244,78,175)(65,245,79,176)(66,246,80,177)(67,247,81,178)(68,248,82,179)(69,249,83,180)(70,250,84,166)(71,251,85,167)(72,252,86,168)(73,253,87,169)(74,254,88,170)(75,255,89,171)(106,427,475,363)(107,428,476,364)(108,429,477,365)(109,430,478,366)(110,431,479,367)(111,432,480,368)(112,433,466,369)(113,434,467,370)(114,435,468,371)(115,421,469,372)(116,422,470,373)(117,423,471,374)(118,424,472,375)(119,425,473,361)(120,426,474,362)(121,217,258,231)(122,218,259,232)(123,219,260,233)(124,220,261,234)(125,221,262,235)(126,222,263,236)(127,223,264,237)(128,224,265,238)(129,225,266,239)(130,211,267,240)(131,212,268,226)(132,213,269,227)(133,214,270,228)(134,215,256,229)(135,216,257,230)(136,393,387,280)(137,394,388,281)(138,395,389,282)(139,396,390,283)(140,397,376,284)(141,398,377,285)(142,399,378,271)(143,400,379,272)(144,401,380,273)(145,402,381,274)(146,403,382,275)(147,404,383,276)(148,405,384,277)(149,391,385,278)(150,392,386,279)(196,318,448,462)(197,319,449,463)(198,320,450,464)(199,321,436,465)(200,322,437,451)(201,323,438,452)(202,324,439,453)(203,325,440,454)(204,326,441,455)(205,327,442,456)(206,328,443,457)(207,329,444,458)(208,330,445,459)(209,316,446,460)(210,317,447,461), (1,437,138,268)(2,438,139,269)(3,439,140,270)(4,440,141,256)(5,441,142,257)(6,442,143,258)(7,443,144,259)(8,444,145,260)(9,445,146,261)(10,446,147,262)(11,447,148,263)(12,448,149,264)(13,449,150,265)(14,450,136,266)(15,436,137,267)(16,185,80,117)(17,186,81,118)(18,187,82,119)(19,188,83,120)(20,189,84,106)(21,190,85,107)(22,191,86,108)(23,192,87,109)(24,193,88,110)(25,194,89,111)(26,195,90,112)(27,181,76,113)(28,182,77,114)(29,183,78,115)(30,184,79,116)(31,66,471,48)(32,67,472,49)(33,68,473,50)(34,69,474,51)(35,70,475,52)(36,71,476,53)(37,72,477,54)(38,73,478,55)(39,74,479,56)(40,75,480,57)(41,61,466,58)(42,62,467,59)(43,63,468,60)(44,64,469,46)(45,65,470,47)(91,242,370,407)(92,243,371,408)(93,244,372,409)(94,245,373,410)(95,246,374,411)(96,247,375,412)(97,248,361,413)(98,249,362,414)(99,250,363,415)(100,251,364,416)(101,252,365,417)(102,253,366,418)(103,254,367,419)(104,255,368,420)(105,241,369,406)(121,300,205,379)(122,286,206,380)(123,287,207,381)(124,288,208,382)(125,289,209,383)(126,290,210,384)(127,291,196,385)(128,292,197,386)(129,293,198,387)(130,294,199,388)(131,295,200,389)(132,296,201,390)(133,297,202,376)(134,298,203,377)(135,299,204,378)(151,344,171,432)(152,345,172,433)(153,331,173,434)(154,332,174,435)(155,333,175,421)(156,334,176,422)(157,335,177,423)(158,336,178,424)(159,337,179,425)(160,338,180,426)(161,339,166,427)(162,340,167,428)(163,341,168,429)(164,342,169,430)(165,343,170,431)(211,353,321,281)(212,354,322,282)(213,355,323,283)(214,356,324,284)(215,357,325,285)(216,358,326,271)(217,359,327,272)(218,360,328,273)(219,346,329,274)(220,347,330,275)(221,348,316,276)(222,349,317,277)(223,350,318,278)(224,351,319,279)(225,352,320,280)(226,309,451,395)(227,310,452,396)(228,311,453,397)(229,312,454,398)(230,313,455,399)(231,314,456,400)(232,315,457,401)(233,301,458,402)(234,302,459,403)(235,303,460,404)(236,304,461,405)(237,305,462,391)(238,306,463,392)(239,307,464,393)(240,308,465,394), (1,253,138,418)(2,254,139,419)(3,255,140,420)(4,241,141,406)(5,242,142,407)(6,243,143,408)(7,244,144,409)(8,245,145,410)(9,246,146,411)(10,247,147,412)(11,248,148,413)(12,249,149,414)(13,250,150,415)(14,251,136,416)(15,252,137,417)(16,347,80,275)(17,348,81,276)(18,349,82,277)(19,350,83,278)(20,351,84,279)(21,352,85,280)(22,353,86,281)(23,354,87,282)(24,355,88,283)(25,356,89,284)(26,357,90,285)(27,358,76,271)(28,359,77,272)(29,360,78,273)(30,346,79,274)(31,234,471,459)(32,235,472,460)(33,236,473,461)(34,237,474,462)(35,238,475,463)(36,239,476,464)(37,240,477,465)(38,226,478,451)(39,227,479,452)(40,228,480,453)(41,229,466,454)(42,230,467,455)(43,231,468,456)(44,232,469,457)(45,233,470,458)(46,315,64,401)(47,301,65,402)(48,302,66,403)(49,303,67,404)(50,304,68,405)(51,305,69,391)(52,306,70,392)(53,307,71,393)(54,308,72,394)(55,309,73,395)(56,310,74,396)(57,311,75,397)(58,312,61,398)(59,313,62,399)(60,314,63,400)(91,257,370,441)(92,258,371,442)(93,259,372,443)(94,260,373,444)(95,261,374,445)(96,262,375,446)(97,263,361,447)(98,264,362,448)(99,265,363,449)(100,266,364,450)(101,267,365,436)(102,268,366,437)(103,269,367,438)(104,270,368,439)(105,256,369,440)(106,319,189,224)(107,320,190,225)(108,321,191,211)(109,322,192,212)(110,323,193,213)(111,324,194,214)(112,325,195,215)(113,326,181,216)(114,327,182,217)(115,328,183,218)(116,329,184,219)(117,330,185,220)(118,316,186,221)(119,317,187,222)(120,318,188,223)(121,435,205,332)(122,421,206,333)(123,422,207,334)(124,423,208,335)(125,424,209,336)(126,425,210,337)(127,426,196,338)(128,427,197,339)(129,428,198,340)(130,429,199,341)(131,430,200,342)(132,431,201,343)(133,432,202,344)(134,433,203,345)(135,434,204,331)(151,297,171,376)(152,298,172,377)(153,299,173,378)(154,300,174,379)(155,286,175,380)(156,287,176,381)(157,288,177,382)(158,289,178,383)(159,290,179,384)(160,291,180,385)(161,292,166,386)(162,293,167,387)(163,294,168,388)(164,295,169,389)(165,296,170,390)>;
G:=Group( (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), (1,309,295,354)(2,310,296,355)(3,311,297,356)(4,312,298,357)(5,313,299,358)(6,314,300,359)(7,315,286,360)(8,301,287,346)(9,302,288,347)(10,303,289,348)(11,304,290,349)(12,305,291,350)(13,306,292,351)(14,307,293,352)(15,308,294,353)(16,157,48,411)(17,158,49,412)(18,159,50,413)(19,160,51,414)(20,161,52,415)(21,162,53,416)(22,163,54,417)(23,164,55,418)(24,165,56,419)(25,151,57,420)(26,152,58,406)(27,153,59,407)(28,154,60,408)(29,155,46,409)(30,156,47,410)(31,95,185,335)(32,96,186,336)(33,97,187,337)(34,98,188,338)(35,99,189,339)(36,100,190,340)(37,101,191,341)(38,102,192,342)(39,103,193,343)(40,104,194,344)(41,105,195,345)(42,91,181,331)(43,92,182,332)(44,93,183,333)(45,94,184,334)(61,241,90,172)(62,242,76,173)(63,243,77,174)(64,244,78,175)(65,245,79,176)(66,246,80,177)(67,247,81,178)(68,248,82,179)(69,249,83,180)(70,250,84,166)(71,251,85,167)(72,252,86,168)(73,253,87,169)(74,254,88,170)(75,255,89,171)(106,427,475,363)(107,428,476,364)(108,429,477,365)(109,430,478,366)(110,431,479,367)(111,432,480,368)(112,433,466,369)(113,434,467,370)(114,435,468,371)(115,421,469,372)(116,422,470,373)(117,423,471,374)(118,424,472,375)(119,425,473,361)(120,426,474,362)(121,217,258,231)(122,218,259,232)(123,219,260,233)(124,220,261,234)(125,221,262,235)(126,222,263,236)(127,223,264,237)(128,224,265,238)(129,225,266,239)(130,211,267,240)(131,212,268,226)(132,213,269,227)(133,214,270,228)(134,215,256,229)(135,216,257,230)(136,393,387,280)(137,394,388,281)(138,395,389,282)(139,396,390,283)(140,397,376,284)(141,398,377,285)(142,399,378,271)(143,400,379,272)(144,401,380,273)(145,402,381,274)(146,403,382,275)(147,404,383,276)(148,405,384,277)(149,391,385,278)(150,392,386,279)(196,318,448,462)(197,319,449,463)(198,320,450,464)(199,321,436,465)(200,322,437,451)(201,323,438,452)(202,324,439,453)(203,325,440,454)(204,326,441,455)(205,327,442,456)(206,328,443,457)(207,329,444,458)(208,330,445,459)(209,316,446,460)(210,317,447,461), (1,437,138,268)(2,438,139,269)(3,439,140,270)(4,440,141,256)(5,441,142,257)(6,442,143,258)(7,443,144,259)(8,444,145,260)(9,445,146,261)(10,446,147,262)(11,447,148,263)(12,448,149,264)(13,449,150,265)(14,450,136,266)(15,436,137,267)(16,185,80,117)(17,186,81,118)(18,187,82,119)(19,188,83,120)(20,189,84,106)(21,190,85,107)(22,191,86,108)(23,192,87,109)(24,193,88,110)(25,194,89,111)(26,195,90,112)(27,181,76,113)(28,182,77,114)(29,183,78,115)(30,184,79,116)(31,66,471,48)(32,67,472,49)(33,68,473,50)(34,69,474,51)(35,70,475,52)(36,71,476,53)(37,72,477,54)(38,73,478,55)(39,74,479,56)(40,75,480,57)(41,61,466,58)(42,62,467,59)(43,63,468,60)(44,64,469,46)(45,65,470,47)(91,242,370,407)(92,243,371,408)(93,244,372,409)(94,245,373,410)(95,246,374,411)(96,247,375,412)(97,248,361,413)(98,249,362,414)(99,250,363,415)(100,251,364,416)(101,252,365,417)(102,253,366,418)(103,254,367,419)(104,255,368,420)(105,241,369,406)(121,300,205,379)(122,286,206,380)(123,287,207,381)(124,288,208,382)(125,289,209,383)(126,290,210,384)(127,291,196,385)(128,292,197,386)(129,293,198,387)(130,294,199,388)(131,295,200,389)(132,296,201,390)(133,297,202,376)(134,298,203,377)(135,299,204,378)(151,344,171,432)(152,345,172,433)(153,331,173,434)(154,332,174,435)(155,333,175,421)(156,334,176,422)(157,335,177,423)(158,336,178,424)(159,337,179,425)(160,338,180,426)(161,339,166,427)(162,340,167,428)(163,341,168,429)(164,342,169,430)(165,343,170,431)(211,353,321,281)(212,354,322,282)(213,355,323,283)(214,356,324,284)(215,357,325,285)(216,358,326,271)(217,359,327,272)(218,360,328,273)(219,346,329,274)(220,347,330,275)(221,348,316,276)(222,349,317,277)(223,350,318,278)(224,351,319,279)(225,352,320,280)(226,309,451,395)(227,310,452,396)(228,311,453,397)(229,312,454,398)(230,313,455,399)(231,314,456,400)(232,315,457,401)(233,301,458,402)(234,302,459,403)(235,303,460,404)(236,304,461,405)(237,305,462,391)(238,306,463,392)(239,307,464,393)(240,308,465,394), (1,253,138,418)(2,254,139,419)(3,255,140,420)(4,241,141,406)(5,242,142,407)(6,243,143,408)(7,244,144,409)(8,245,145,410)(9,246,146,411)(10,247,147,412)(11,248,148,413)(12,249,149,414)(13,250,150,415)(14,251,136,416)(15,252,137,417)(16,347,80,275)(17,348,81,276)(18,349,82,277)(19,350,83,278)(20,351,84,279)(21,352,85,280)(22,353,86,281)(23,354,87,282)(24,355,88,283)(25,356,89,284)(26,357,90,285)(27,358,76,271)(28,359,77,272)(29,360,78,273)(30,346,79,274)(31,234,471,459)(32,235,472,460)(33,236,473,461)(34,237,474,462)(35,238,475,463)(36,239,476,464)(37,240,477,465)(38,226,478,451)(39,227,479,452)(40,228,480,453)(41,229,466,454)(42,230,467,455)(43,231,468,456)(44,232,469,457)(45,233,470,458)(46,315,64,401)(47,301,65,402)(48,302,66,403)(49,303,67,404)(50,304,68,405)(51,305,69,391)(52,306,70,392)(53,307,71,393)(54,308,72,394)(55,309,73,395)(56,310,74,396)(57,311,75,397)(58,312,61,398)(59,313,62,399)(60,314,63,400)(91,257,370,441)(92,258,371,442)(93,259,372,443)(94,260,373,444)(95,261,374,445)(96,262,375,446)(97,263,361,447)(98,264,362,448)(99,265,363,449)(100,266,364,450)(101,267,365,436)(102,268,366,437)(103,269,367,438)(104,270,368,439)(105,256,369,440)(106,319,189,224)(107,320,190,225)(108,321,191,211)(109,322,192,212)(110,323,193,213)(111,324,194,214)(112,325,195,215)(113,326,181,216)(114,327,182,217)(115,328,183,218)(116,329,184,219)(117,330,185,220)(118,316,186,221)(119,317,187,222)(120,318,188,223)(121,435,205,332)(122,421,206,333)(123,422,207,334)(124,423,208,335)(125,424,209,336)(126,425,210,337)(127,426,196,338)(128,427,197,339)(129,428,198,340)(130,429,199,341)(131,430,200,342)(132,431,201,343)(133,432,202,344)(134,433,203,345)(135,434,204,331)(151,297,171,376)(152,298,172,377)(153,299,173,378)(154,300,174,379)(155,286,175,380)(156,287,176,381)(157,288,177,382)(158,289,178,383)(159,290,179,384)(160,291,180,385)(161,292,166,386)(162,293,167,387)(163,294,168,388)(164,295,169,389)(165,296,170,390) );
G=PermutationGroup([[(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)], [(1,309,295,354),(2,310,296,355),(3,311,297,356),(4,312,298,357),(5,313,299,358),(6,314,300,359),(7,315,286,360),(8,301,287,346),(9,302,288,347),(10,303,289,348),(11,304,290,349),(12,305,291,350),(13,306,292,351),(14,307,293,352),(15,308,294,353),(16,157,48,411),(17,158,49,412),(18,159,50,413),(19,160,51,414),(20,161,52,415),(21,162,53,416),(22,163,54,417),(23,164,55,418),(24,165,56,419),(25,151,57,420),(26,152,58,406),(27,153,59,407),(28,154,60,408),(29,155,46,409),(30,156,47,410),(31,95,185,335),(32,96,186,336),(33,97,187,337),(34,98,188,338),(35,99,189,339),(36,100,190,340),(37,101,191,341),(38,102,192,342),(39,103,193,343),(40,104,194,344),(41,105,195,345),(42,91,181,331),(43,92,182,332),(44,93,183,333),(45,94,184,334),(61,241,90,172),(62,242,76,173),(63,243,77,174),(64,244,78,175),(65,245,79,176),(66,246,80,177),(67,247,81,178),(68,248,82,179),(69,249,83,180),(70,250,84,166),(71,251,85,167),(72,252,86,168),(73,253,87,169),(74,254,88,170),(75,255,89,171),(106,427,475,363),(107,428,476,364),(108,429,477,365),(109,430,478,366),(110,431,479,367),(111,432,480,368),(112,433,466,369),(113,434,467,370),(114,435,468,371),(115,421,469,372),(116,422,470,373),(117,423,471,374),(118,424,472,375),(119,425,473,361),(120,426,474,362),(121,217,258,231),(122,218,259,232),(123,219,260,233),(124,220,261,234),(125,221,262,235),(126,222,263,236),(127,223,264,237),(128,224,265,238),(129,225,266,239),(130,211,267,240),(131,212,268,226),(132,213,269,227),(133,214,270,228),(134,215,256,229),(135,216,257,230),(136,393,387,280),(137,394,388,281),(138,395,389,282),(139,396,390,283),(140,397,376,284),(141,398,377,285),(142,399,378,271),(143,400,379,272),(144,401,380,273),(145,402,381,274),(146,403,382,275),(147,404,383,276),(148,405,384,277),(149,391,385,278),(150,392,386,279),(196,318,448,462),(197,319,449,463),(198,320,450,464),(199,321,436,465),(200,322,437,451),(201,323,438,452),(202,324,439,453),(203,325,440,454),(204,326,441,455),(205,327,442,456),(206,328,443,457),(207,329,444,458),(208,330,445,459),(209,316,446,460),(210,317,447,461)], [(1,437,138,268),(2,438,139,269),(3,439,140,270),(4,440,141,256),(5,441,142,257),(6,442,143,258),(7,443,144,259),(8,444,145,260),(9,445,146,261),(10,446,147,262),(11,447,148,263),(12,448,149,264),(13,449,150,265),(14,450,136,266),(15,436,137,267),(16,185,80,117),(17,186,81,118),(18,187,82,119),(19,188,83,120),(20,189,84,106),(21,190,85,107),(22,191,86,108),(23,192,87,109),(24,193,88,110),(25,194,89,111),(26,195,90,112),(27,181,76,113),(28,182,77,114),(29,183,78,115),(30,184,79,116),(31,66,471,48),(32,67,472,49),(33,68,473,50),(34,69,474,51),(35,70,475,52),(36,71,476,53),(37,72,477,54),(38,73,478,55),(39,74,479,56),(40,75,480,57),(41,61,466,58),(42,62,467,59),(43,63,468,60),(44,64,469,46),(45,65,470,47),(91,242,370,407),(92,243,371,408),(93,244,372,409),(94,245,373,410),(95,246,374,411),(96,247,375,412),(97,248,361,413),(98,249,362,414),(99,250,363,415),(100,251,364,416),(101,252,365,417),(102,253,366,418),(103,254,367,419),(104,255,368,420),(105,241,369,406),(121,300,205,379),(122,286,206,380),(123,287,207,381),(124,288,208,382),(125,289,209,383),(126,290,210,384),(127,291,196,385),(128,292,197,386),(129,293,198,387),(130,294,199,388),(131,295,200,389),(132,296,201,390),(133,297,202,376),(134,298,203,377),(135,299,204,378),(151,344,171,432),(152,345,172,433),(153,331,173,434),(154,332,174,435),(155,333,175,421),(156,334,176,422),(157,335,177,423),(158,336,178,424),(159,337,179,425),(160,338,180,426),(161,339,166,427),(162,340,167,428),(163,341,168,429),(164,342,169,430),(165,343,170,431),(211,353,321,281),(212,354,322,282),(213,355,323,283),(214,356,324,284),(215,357,325,285),(216,358,326,271),(217,359,327,272),(218,360,328,273),(219,346,329,274),(220,347,330,275),(221,348,316,276),(222,349,317,277),(223,350,318,278),(224,351,319,279),(225,352,320,280),(226,309,451,395),(227,310,452,396),(228,311,453,397),(229,312,454,398),(230,313,455,399),(231,314,456,400),(232,315,457,401),(233,301,458,402),(234,302,459,403),(235,303,460,404),(236,304,461,405),(237,305,462,391),(238,306,463,392),(239,307,464,393),(240,308,465,394)], [(1,253,138,418),(2,254,139,419),(3,255,140,420),(4,241,141,406),(5,242,142,407),(6,243,143,408),(7,244,144,409),(8,245,145,410),(9,246,146,411),(10,247,147,412),(11,248,148,413),(12,249,149,414),(13,250,150,415),(14,251,136,416),(15,252,137,417),(16,347,80,275),(17,348,81,276),(18,349,82,277),(19,350,83,278),(20,351,84,279),(21,352,85,280),(22,353,86,281),(23,354,87,282),(24,355,88,283),(25,356,89,284),(26,357,90,285),(27,358,76,271),(28,359,77,272),(29,360,78,273),(30,346,79,274),(31,234,471,459),(32,235,472,460),(33,236,473,461),(34,237,474,462),(35,238,475,463),(36,239,476,464),(37,240,477,465),(38,226,478,451),(39,227,479,452),(40,228,480,453),(41,229,466,454),(42,230,467,455),(43,231,468,456),(44,232,469,457),(45,233,470,458),(46,315,64,401),(47,301,65,402),(48,302,66,403),(49,303,67,404),(50,304,68,405),(51,305,69,391),(52,306,70,392),(53,307,71,393),(54,308,72,394),(55,309,73,395),(56,310,74,396),(57,311,75,397),(58,312,61,398),(59,313,62,399),(60,314,63,400),(91,257,370,441),(92,258,371,442),(93,259,372,443),(94,260,373,444),(95,261,374,445),(96,262,375,446),(97,263,361,447),(98,264,362,448),(99,265,363,449),(100,266,364,450),(101,267,365,436),(102,268,366,437),(103,269,367,438),(104,270,368,439),(105,256,369,440),(106,319,189,224),(107,320,190,225),(108,321,191,211),(109,322,192,212),(110,323,193,213),(111,324,194,214),(112,325,195,215),(113,326,181,216),(114,327,182,217),(115,328,183,218),(116,329,184,219),(117,330,185,220),(118,316,186,221),(119,317,187,222),(120,318,188,223),(121,435,205,332),(122,421,206,333),(123,422,207,334),(124,423,208,335),(125,424,209,336),(126,425,210,337),(127,426,196,338),(128,427,197,339),(129,428,198,340),(130,429,199,341),(131,430,200,342),(132,431,201,343),(133,432,202,344),(134,433,203,345),(135,434,204,331),(151,297,171,376),(152,298,172,377),(153,299,173,378),(154,300,174,379),(155,286,175,380),(156,287,176,381),(157,288,177,382),(158,289,178,383),(159,290,179,384),(160,291,180,385),(161,292,166,386),(162,293,167,387),(163,294,168,388),(164,295,169,389),(165,296,170,390)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 10A | ··· | 10L | 12A | ··· | 12L | 12M | ··· | 12T | 15A | ··· | 15H | 20A | ··· | 20X | 20Y | ··· | 20AN | 30A | ··· | 30X | 60A | ··· | 60AV | 60AW | ··· | 60CB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | D4 | Q8 | C3×D4 | C3×Q8 | C5×D4 | C5×Q8 | D4×C15 | Q8×C15 |
| kernel | C15×C4⋊Q8 | C4×C60 | C15×C4⋊C4 | Q8×C30 | C5×C4⋊Q8 | C3×C4⋊Q8 | C4×C20 | C5×C4⋊C4 | Q8×C10 | C4×C12 | C3×C4⋊C4 | C6×Q8 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C60 | C60 | C20 | C20 | C12 | C12 | C4 | C4 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 | 8 | 8 | 8 | 32 | 16 | 2 | 4 | 4 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C15×C4⋊Q8 ►in GL4(𝔽61) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 42 |
| 0 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 20 | 59 |
| 0 | 0 | 48 | 41 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 16 | 32 | 0 | 0 |
| 32 | 45 | 0 | 0 |
| 0 | 0 | 4 | 47 |
| 0 | 0 | 49 | 57 |
G:=sub<GL(4,GF(61))| [15,0,0,0,0,15,0,0,0,0,42,0,0,0,0,42],[0,1,0,0,60,0,0,0,0,0,20,48,0,0,59,41],[0,60,0,0,1,0,0,0,0,0,60,0,0,0,0,60],[16,32,0,0,32,45,0,0,0,0,4,49,0,0,47,57] >;
C15×C4⋊Q8 in GAP, Magma, Sage, TeX
C_{15}\times C_4\rtimes Q_8 % in TeX
G:=Group("C15xC4:Q8"); // GroupNames label
G:=SmallGroup(480,933);
// by ID
G=gap.SmallGroup(480,933);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,840,1709,848,5126,1276]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations