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## G = C15×Q8⋊C4order 480 = 25·3·5

### Direct product of C15 and Q8⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C15×Q8⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — C2×C60 — C15×C4⋊C4 — C15×Q8⋊C4
 Lower central C1 — C2 — C4 — C15×Q8⋊C4
 Upper central C1 — C2×C30 — C2×C60 — C15×Q8⋊C4

Generators and relations for C15×Q8⋊C4
G = < a,b,c,d | a15=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 120 in 84 conjugacy classes, 56 normal (48 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, Q8, Q8, C10, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, C20, C20, C2×C10, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, C30, Q8⋊C4, C40, C2×C20, C2×C20, C5×Q8, C5×Q8, C3×C4⋊C4, C2×C24, C6×Q8, C60, C60, C2×C30, C5×C4⋊C4, C2×C40, Q8×C10, C3×Q8⋊C4, C120, C2×C60, C2×C60, Q8×C15, Q8×C15, C5×Q8⋊C4, C15×C4⋊C4, C2×C120, Q8×C30, C15×Q8⋊C4
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, C10, C12, C2×C6, C15, C22⋊C4, SD16, Q16, C20, C2×C10, C2×C12, C3×D4, C30, Q8⋊C4, C2×C20, C5×D4, C3×C22⋊C4, C3×SD16, C3×Q16, C60, C2×C30, C5×C22⋊C4, C5×SD16, C5×Q16, C3×Q8⋊C4, C2×C60, D4×C15, C5×Q8⋊C4, C15×C22⋊C4, C15×SD16, C15×Q16, C15×Q8⋊C4

Smallest permutation representation of C15×Q8⋊C4
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)
(1 369 201 331)(2 370 202 332)(3 371 203 333)(4 372 204 334)(5 373 205 335)(6 374 206 336)(7 375 207 337)(8 361 208 338)(9 362 209 339)(10 363 210 340)(11 364 196 341)(12 365 197 342)(13 366 198 343)(14 367 199 344)(15 368 200 345)(16 105 141 433)(17 91 142 434)(18 92 143 435)(19 93 144 421)(20 94 145 422)(21 95 146 423)(22 96 147 424)(23 97 148 425)(24 98 149 426)(25 99 150 427)(26 100 136 428)(27 101 137 429)(28 102 138 430)(29 103 139 431)(30 104 140 432)(31 360 188 316)(32 346 189 317)(33 347 190 318)(34 348 191 319)(35 349 192 320)(36 350 193 321)(37 351 194 322)(38 352 195 323)(39 353 181 324)(40 354 182 325)(41 355 183 326)(42 356 184 327)(43 357 185 328)(44 358 186 329)(45 359 187 330)(46 275 129 173)(47 276 130 174)(48 277 131 175)(49 278 132 176)(50 279 133 177)(51 280 134 178)(52 281 135 179)(53 282 121 180)(54 283 122 166)(55 284 123 167)(56 285 124 168)(57 271 125 169)(58 272 126 170)(59 273 127 171)(60 274 128 172)(61 237 441 314)(62 238 442 315)(63 239 443 301)(64 240 444 302)(65 226 445 303)(66 227 446 304)(67 228 447 305)(68 229 448 306)(69 230 449 307)(70 231 450 308)(71 232 436 309)(72 233 437 310)(73 234 438 311)(74 235 439 312)(75 236 440 313)(76 162 476 108)(77 163 477 109)(78 164 478 110)(79 165 479 111)(80 151 480 112)(81 152 466 113)(82 153 467 114)(83 154 468 115)(84 155 469 116)(85 156 470 117)(86 157 471 118)(87 158 472 119)(88 159 473 120)(89 160 474 106)(90 161 475 107)(211 265 252 411)(212 266 253 412)(213 267 254 413)(214 268 255 414)(215 269 241 415)(216 270 242 416)(217 256 243 417)(218 257 244 418)(219 258 245 419)(220 259 246 420)(221 260 247 406)(222 261 248 407)(223 262 249 408)(224 263 250 409)(225 264 251 410)(286 398 382 456)(287 399 383 457)(288 400 384 458)(289 401 385 459)(290 402 386 460)(291 403 387 461)(292 404 388 462)(293 405 389 463)(294 391 390 464)(295 392 376 465)(296 393 377 451)(297 394 378 452)(298 395 379 453)(299 396 380 454)(300 397 381 455)
(1 97 201 425)(2 98 202 426)(3 99 203 427)(4 100 204 428)(5 101 205 429)(6 102 206 430)(7 103 207 431)(8 104 208 432)(9 105 209 433)(10 91 210 434)(11 92 196 435)(12 93 197 421)(13 94 198 422)(14 95 199 423)(15 96 200 424)(16 339 141 362)(17 340 142 363)(18 341 143 364)(19 342 144 365)(20 343 145 366)(21 344 146 367)(22 345 147 368)(23 331 148 369)(24 332 149 370)(25 333 150 371)(26 334 136 372)(27 335 137 373)(28 336 138 374)(29 337 139 375)(30 338 140 361)(31 232 188 309)(32 233 189 310)(33 234 190 311)(34 235 191 312)(35 236 192 313)(36 237 193 314)(37 238 194 315)(38 239 195 301)(39 240 181 302)(40 226 182 303)(41 227 183 304)(42 228 184 305)(43 229 185 306)(44 230 186 307)(45 231 187 308)(46 466 129 81)(47 467 130 82)(48 468 131 83)(49 469 132 84)(50 470 133 85)(51 471 134 86)(52 472 135 87)(53 473 121 88)(54 474 122 89)(55 475 123 90)(56 476 124 76)(57 477 125 77)(58 478 126 78)(59 479 127 79)(60 480 128 80)(61 321 441 350)(62 322 442 351)(63 323 443 352)(64 324 444 353)(65 325 445 354)(66 326 446 355)(67 327 447 356)(68 328 448 357)(69 329 449 358)(70 330 450 359)(71 316 436 360)(72 317 437 346)(73 318 438 347)(74 319 439 348)(75 320 440 349)(106 283 160 166)(107 284 161 167)(108 285 162 168)(109 271 163 169)(110 272 164 170)(111 273 165 171)(112 274 151 172)(113 275 152 173)(114 276 153 174)(115 277 154 175)(116 278 155 176)(117 279 156 177)(118 280 157 178)(119 281 158 179)(120 282 159 180)(211 287 252 383)(212 288 253 384)(213 289 254 385)(214 290 255 386)(215 291 241 387)(216 292 242 388)(217 293 243 389)(218 294 244 390)(219 295 245 376)(220 296 246 377)(221 297 247 378)(222 298 248 379)(223 299 249 380)(224 300 250 381)(225 286 251 382)(256 463 417 405)(257 464 418 391)(258 465 419 392)(259 451 420 393)(260 452 406 394)(261 453 407 395)(262 454 408 396)(263 455 409 397)(264 456 410 398)(265 457 411 399)(266 458 412 400)(267 459 413 401)(268 460 414 402)(269 461 415 403)(270 462 416 404)
(1 468 326 221)(2 469 327 222)(3 470 328 223)(4 471 329 224)(5 472 330 225)(6 473 316 211)(7 474 317 212)(8 475 318 213)(9 476 319 214)(10 477 320 215)(11 478 321 216)(12 479 322 217)(13 480 323 218)(14 466 324 219)(15 467 325 220)(16 56 235 386)(17 57 236 387)(18 58 237 388)(19 59 238 389)(20 60 239 390)(21 46 240 376)(22 47 226 377)(23 48 227 378)(24 49 228 379)(25 50 229 380)(26 51 230 381)(27 52 231 382)(28 53 232 383)(29 54 233 384)(30 55 234 385)(31 411 374 159)(32 412 375 160)(33 413 361 161)(34 414 362 162)(35 415 363 163)(36 416 364 164)(37 417 365 165)(38 418 366 151)(39 419 367 152)(40 420 368 153)(41 406 369 154)(42 407 370 155)(43 408 371 156)(44 409 372 157)(45 410 373 158)(61 462 435 272)(62 463 421 273)(63 464 422 274)(64 465 423 275)(65 451 424 276)(66 452 425 277)(67 453 426 278)(68 454 427 279)(69 455 428 280)(70 456 429 281)(71 457 430 282)(72 458 431 283)(73 459 432 284)(74 460 433 285)(75 461 434 271)(76 348 255 209)(77 349 241 210)(78 350 242 196)(79 351 243 197)(80 352 244 198)(81 353 245 199)(82 354 246 200)(83 355 247 201)(84 356 248 202)(85 357 249 203)(86 358 250 204)(87 359 251 205)(88 360 252 206)(89 346 253 207)(90 347 254 208)(91 169 440 403)(92 170 441 404)(93 171 442 405)(94 172 443 391)(95 173 444 392)(96 174 445 393)(97 175 446 394)(98 176 447 395)(99 177 448 396)(100 178 449 397)(101 179 450 398)(102 180 436 399)(103 166 437 400)(104 167 438 401)(105 168 439 402)(106 189 266 337)(107 190 267 338)(108 191 268 339)(109 192 269 340)(110 193 270 341)(111 194 256 342)(112 195 257 343)(113 181 258 344)(114 182 259 345)(115 183 260 331)(116 184 261 332)(117 185 262 333)(118 186 263 334)(119 187 264 335)(120 188 265 336)(121 309 287 138)(122 310 288 139)(123 311 289 140)(124 312 290 141)(125 313 291 142)(126 314 292 143)(127 315 293 144)(128 301 294 145)(129 302 295 146)(130 303 296 147)(131 304 297 148)(132 305 298 149)(133 306 299 150)(134 307 300 136)(135 308 286 137)

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

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

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

210 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A ··· 6F 8A 8B 8C 8D 10A ··· 10L 12A 12B 12C 12D 12E ··· 12L 15A ··· 15H 20A ··· 20H 20I ··· 20X 24A ··· 24H 30A ··· 30X 40A ··· 40P 60A ··· 60P 60Q ··· 60AV 120A ··· 120AF order 1 2 2 2 3 3 4 4 4 4 4 4 5 5 5 5 6 ··· 6 8 8 8 8 10 ··· 10 12 12 12 12 12 ··· 12 15 ··· 15 20 ··· 20 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 120 ··· 120 size 1 1 1 1 1 1 2 2 4 4 4 4 1 1 1 1 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2

210 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C3 C4 C5 C6 C6 C6 C10 C10 C10 C12 C15 C20 C30 C30 C30 C60 D4 D4 SD16 Q16 C3×D4 C3×D4 C5×D4 C5×D4 C3×SD16 C3×Q16 C5×SD16 C5×Q16 D4×C15 D4×C15 C15×SD16 C15×Q16 kernel C15×Q8⋊C4 C15×C4⋊C4 C2×C120 Q8×C30 C5×Q8⋊C4 Q8×C15 C3×Q8⋊C4 C5×C4⋊C4 C2×C40 Q8×C10 C3×C4⋊C4 C2×C24 C6×Q8 C5×Q8 Q8⋊C4 C3×Q8 C4⋊C4 C2×C8 C2×Q8 Q8 C60 C2×C30 C30 C30 C20 C2×C10 C12 C2×C6 C10 C10 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 2 4 4 2 2 2 4 4 4 8 8 16 8 8 8 32 1 1 2 2 2 2 4 4 4 4 8 8 8 8 16 16

Matrix representation of C15×Q8⋊C4 in GL4(𝔽241) generated by

 1 0 0 0 0 225 0 0 0 0 87 0 0 0 0 87
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 240 0
,
 240 0 0 0 0 240 0 0 0 0 3 58 0 0 58 238
,
 64 0 0 0 0 1 0 0 0 0 52 118 0 0 118 189
G:=sub<GL(4,GF(241))| [1,0,0,0,0,225,0,0,0,0,87,0,0,0,0,87],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,0],[240,0,0,0,0,240,0,0,0,0,3,58,0,0,58,238],[64,0,0,0,0,1,0,0,0,0,52,118,0,0,118,189] >;

C15×Q8⋊C4 in GAP, Magma, Sage, TeX

C_{15}\times Q_8\rtimes C_4
% in TeX

G:=Group("C15xQ8:C4");
// GroupNames label

G:=SmallGroup(480,206);
// by ID

G=gap.SmallGroup(480,206);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,1688,10504,5261,172]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations

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