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## G = C15×C4⋊Q8order 480 = 25·3·5

### Direct product of C15 and C4⋊Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C15×C4⋊Q8
 Chief series C1 — C2 — C22 — C2×C10 — C2×C30 — C2×C60 — Q8×C30 — C15×C4⋊Q8
 Lower central C1 — C22 — C15×C4⋊Q8
 Upper central C1 — C2×C30 — C15×C4⋊Q8

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 [×2], C3, C4 [×6], C4 [×4], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×6], Q8 [×4], C10, C10 [×2], C12 [×6], C12 [×4], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], C20 [×6], C20 [×4], C2×C10, C2×C12, C2×C12 [×6], C3×Q8 [×4], C30, C30 [×2], C4⋊Q8, C2×C20, C2×C20 [×6], C5×Q8 [×4], C4×C12, C3×C4⋊C4 [×4], C6×Q8 [×2], C60 [×6], C60 [×4], C2×C30, C4×C20, C5×C4⋊C4 [×4], Q8×C10 [×2], C3×C4⋊Q8, C2×C60, C2×C60 [×6], Q8×C15 [×4], C5×C4⋊Q8, C4×C60, C15×C4⋊C4 [×4], Q8×C30 [×2], C15×C4⋊Q8
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], D4 [×2], Q8 [×4], C23, C10 [×7], C2×C6 [×7], C15, C2×D4, C2×Q8 [×2], C2×C10 [×7], C3×D4 [×2], C3×Q8 [×4], C22×C6, C30 [×7], C4⋊Q8, C5×D4 [×2], C5×Q8 [×4], C22×C10, C6×D4, C6×Q8 [×2], C2×C30 [×7], D4×C10, Q8×C10 [×2], C3×C4⋊Q8, D4×C15 [×2], Q8×C15 [×4], C22×C30, C5×C4⋊Q8, D4×C30, Q8×C30 [×2], C15×C4⋊Q8

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

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

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

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

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

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