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G = C4×C15⋊Q8order 480 = 25·3·5

Direct product of C4 and C15⋊Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×C15⋊Q8, C605Q8, C205Dic6, C125Dic10, C1512(C4×Q8), C52(C4×Dic6), C31(C4×Dic10), C30.55(C2×Q8), (C2×C20).342D6, Dic3.8(C4×D5), C30.97(C4○D4), C6.43(C4○D20), (C2×C12).346D10, C30.66(C22×C4), (C4×Dic5).12S3, Dic5.16(C4×S3), (C12×Dic5).9C2, (C4×Dic3).10D5, (Dic3×C20).9C2, C10.23(C2×Dic6), C6.22(C2×Dic10), C10.46(C4○D12), (C2×C60).244C22, (C2×C30).157C23, (C4×Dic15).20C2, (C2×Dic5).186D6, Dic15.40(C2×C4), C30.Q8.17C2, C6.Dic10.17C2, Dic155C4.17C2, (C2×Dic3).159D10, C2.7(D6.D10), (C6×Dic5).213C22, (C10×Dic3).193C22, (C2×Dic15).218C22, C2.36(C4×S3×D5), C6.34(C2×C4×D5), C2.1(C2×C15⋊Q8), C10.67(S3×C2×C4), (C2×C15⋊Q8).11C2, C22.69(C2×S3×D5), (C2×C4).247(S3×D5), (C5×Dic3).36(C2×C4), (C3×Dic5).42(C2×C4), (C2×C6).169(C22×D5), (C2×C10).169(C22×S3), SmallGroup(480,543)

Series: Derived Chief Lower central Upper central

C1C30 — C4×C15⋊Q8
C1C5C15C30C2×C30C6×Dic5C2×C15⋊Q8 — C4×C15⋊Q8
C15C30 — C4×C15⋊Q8
C1C2×C4

Generators and relations for C4×C15⋊Q8
 G = < a,b,c,d | a4=b15=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b11, dbd-1=b4, dcd-1=c-1 >

Subgroups: 556 in 140 conjugacy classes, 66 normal (44 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×9], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×3], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×3], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4×Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15 [×2], Dic15, C60 [×2], C2×C30, C4×Dic5, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C4×C20, C2×Dic10, C4×Dic6, C15⋊Q8 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, C4×Dic10, C30.Q8, Dic155C4, C6.Dic10, C12×Dic5, Dic3×C20, C4×Dic15, C2×C15⋊Q8, C4×C15⋊Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D5, D6 [×3], C22×C4, C2×Q8, C4○D4, D10 [×3], Dic6 [×2], C4×S3 [×2], C22×S3, C4×Q8, Dic10 [×2], C4×D5 [×2], C22×D5, C2×Dic6, S3×C2×C4, C4○D12, S3×D5, C2×Dic10, C2×C4×D5, C4○D20, C4×Dic6, C15⋊Q8 [×2], C2×S3×D5, C4×Dic10, D6.D10, C4×S3×D5, C2×C15⋊Q8, C4×C15⋊Q8

Smallest permutation representation of C4×C15⋊Q8
Regular action on 480 points
Generators in S480
(1 95 44 71)(2 96 45 72)(3 97 31 73)(4 98 32 74)(5 99 33 75)(6 100 34 61)(7 101 35 62)(8 102 36 63)(9 103 37 64)(10 104 38 65)(11 105 39 66)(12 91 40 67)(13 92 41 68)(14 93 42 69)(15 94 43 70)(16 118 47 88)(17 119 48 89)(18 120 49 90)(19 106 50 76)(20 107 51 77)(21 108 52 78)(22 109 53 79)(23 110 54 80)(24 111 55 81)(25 112 56 82)(26 113 57 83)(27 114 58 84)(28 115 59 85)(29 116 60 86)(30 117 46 87)(121 215 155 185)(122 216 156 186)(123 217 157 187)(124 218 158 188)(125 219 159 189)(126 220 160 190)(127 221 161 191)(128 222 162 192)(129 223 163 193)(130 224 164 194)(131 225 165 195)(132 211 151 181)(133 212 152 182)(134 213 153 183)(135 214 154 184)(136 237 166 196)(137 238 167 197)(138 239 168 198)(139 240 169 199)(140 226 170 200)(141 227 171 201)(142 228 172 202)(143 229 173 203)(144 230 174 204)(145 231 175 205)(146 232 176 206)(147 233 177 207)(148 234 178 208)(149 235 179 209)(150 236 180 210)(241 335 278 311)(242 336 279 312)(243 337 280 313)(244 338 281 314)(245 339 282 315)(246 340 283 301)(247 341 284 302)(248 342 285 303)(249 343 271 304)(250 344 272 305)(251 345 273 306)(252 331 274 307)(253 332 275 308)(254 333 276 309)(255 334 277 310)(256 349 296 329)(257 350 297 330)(258 351 298 316)(259 352 299 317)(260 353 300 318)(261 354 286 319)(262 355 287 320)(263 356 288 321)(264 357 289 322)(265 358 290 323)(266 359 291 324)(267 360 292 325)(268 346 293 326)(269 347 294 327)(270 348 295 328)(361 463 391 432)(362 464 392 433)(363 465 393 434)(364 451 394 435)(365 452 395 421)(366 453 396 422)(367 454 397 423)(368 455 398 424)(369 456 399 425)(370 457 400 426)(371 458 401 427)(372 459 402 428)(373 460 403 429)(374 461 404 430)(375 462 405 431)(376 470 410 447)(377 471 411 448)(378 472 412 449)(379 473 413 450)(380 474 414 436)(381 475 415 437)(382 476 416 438)(383 477 417 439)(384 478 418 440)(385 479 419 441)(386 480 420 442)(387 466 406 443)(388 467 407 444)(389 468 408 445)(390 469 409 446)
(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)
(1 147 30 130)(2 143 16 126)(3 139 17 122)(4 150 18 133)(5 146 19 129)(6 142 20 125)(7 138 21 121)(8 149 22 132)(9 145 23 128)(10 141 24 124)(11 137 25 135)(12 148 26 131)(13 144 27 127)(14 140 28 123)(15 136 29 134)(31 169 48 156)(32 180 49 152)(33 176 50 163)(34 172 51 159)(35 168 52 155)(36 179 53 151)(37 175 54 162)(38 171 55 158)(39 167 56 154)(40 178 57 165)(41 174 58 161)(42 170 59 157)(43 166 60 153)(44 177 46 164)(45 173 47 160)(61 202 77 189)(62 198 78 185)(63 209 79 181)(64 205 80 192)(65 201 81 188)(66 197 82 184)(67 208 83 195)(68 204 84 191)(69 200 85 187)(70 196 86 183)(71 207 87 194)(72 203 88 190)(73 199 89 186)(74 210 90 182)(75 206 76 193)(91 234 113 225)(92 230 114 221)(93 226 115 217)(94 237 116 213)(95 233 117 224)(96 229 118 220)(97 240 119 216)(98 236 120 212)(99 232 106 223)(100 228 107 219)(101 239 108 215)(102 235 109 211)(103 231 110 222)(104 227 111 218)(105 238 112 214)(241 362 265 381)(242 373 266 377)(243 369 267 388)(244 365 268 384)(245 361 269 380)(246 372 270 376)(247 368 256 387)(248 364 257 383)(249 375 258 379)(250 371 259 390)(251 367 260 386)(252 363 261 382)(253 374 262 378)(254 370 263 389)(255 366 264 385)(271 405 298 413)(272 401 299 409)(273 397 300 420)(274 393 286 416)(275 404 287 412)(276 400 288 408)(277 396 289 419)(278 392 290 415)(279 403 291 411)(280 399 292 407)(281 395 293 418)(282 391 294 414)(283 402 295 410)(284 398 296 406)(285 394 297 417)(301 428 328 447)(302 424 329 443)(303 435 330 439)(304 431 316 450)(305 427 317 446)(306 423 318 442)(307 434 319 438)(308 430 320 449)(309 426 321 445)(310 422 322 441)(311 433 323 437)(312 429 324 448)(313 425 325 444)(314 421 326 440)(315 432 327 436)(331 465 354 476)(332 461 355 472)(333 457 356 468)(334 453 357 479)(335 464 358 475)(336 460 359 471)(337 456 360 467)(338 452 346 478)(339 463 347 474)(340 459 348 470)(341 455 349 466)(342 451 350 477)(343 462 351 473)(344 458 352 469)(345 454 353 480)
(1 268 30 244)(2 257 16 248)(3 261 17 252)(4 265 18 241)(5 269 19 245)(6 258 20 249)(7 262 21 253)(8 266 22 242)(9 270 23 246)(10 259 24 250)(11 263 25 254)(12 267 26 243)(13 256 27 247)(14 260 28 251)(15 264 29 255)(31 286 48 274)(32 290 49 278)(33 294 50 282)(34 298 51 271)(35 287 52 275)(36 291 53 279)(37 295 54 283)(38 299 55 272)(39 288 56 276)(40 292 57 280)(41 296 58 284)(42 300 59 273)(43 289 60 277)(44 293 46 281)(45 297 47 285)(61 316 77 304)(62 320 78 308)(63 324 79 312)(64 328 80 301)(65 317 81 305)(66 321 82 309)(67 325 83 313)(68 329 84 302)(69 318 85 306)(70 322 86 310)(71 326 87 314)(72 330 88 303)(73 319 89 307)(74 323 90 311)(75 327 76 315)(91 360 113 337)(92 349 114 341)(93 353 115 345)(94 357 116 334)(95 346 117 338)(96 350 118 342)(97 354 119 331)(98 358 120 335)(99 347 106 339)(100 351 107 343)(101 355 108 332)(102 359 109 336)(103 348 110 340)(104 352 111 344)(105 356 112 333)(121 378 138 374)(122 382 139 363)(123 386 140 367)(124 390 141 371)(125 379 142 375)(126 383 143 364)(127 387 144 368)(128 376 145 372)(129 380 146 361)(130 384 147 365)(131 388 148 369)(132 377 149 373)(133 381 150 362)(134 385 136 366)(135 389 137 370)(151 411 179 403)(152 415 180 392)(153 419 166 396)(154 408 167 400)(155 412 168 404)(156 416 169 393)(157 420 170 397)(158 409 171 401)(159 413 172 405)(160 417 173 394)(161 406 174 398)(162 410 175 402)(163 414 176 391)(164 418 177 395)(165 407 178 399)(181 448 209 429)(182 437 210 433)(183 441 196 422)(184 445 197 426)(185 449 198 430)(186 438 199 434)(187 442 200 423)(188 446 201 427)(189 450 202 431)(190 439 203 435)(191 443 204 424)(192 447 205 428)(193 436 206 432)(194 440 207 421)(195 444 208 425)(211 471 235 460)(212 475 236 464)(213 479 237 453)(214 468 238 457)(215 472 239 461)(216 476 240 465)(217 480 226 454)(218 469 227 458)(219 473 228 462)(220 477 229 451)(221 466 230 455)(222 470 231 459)(223 474 232 463)(224 478 233 452)(225 467 234 456)

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B6A6B6C10A···10F12A12B12C12D12E···12L15A15B20A···20H20I···20X30A···30F60A···60H
order1222344444444444444445566610···101212121212···12151520···2020···2030···3060···60
size11112111166661010101030303030222222···2222210···10442···26···64···44···4

84 irreducible representations

dim1111111112222222222222244444
type+++++++++-+++++--+-+
imageC1C2C2C2C2C2C2C2C4S3Q8D5D6D6C4○D4D10D10C4×S3Dic6C4×D5Dic10C4○D12C4○D20S3×D5C15⋊Q8C2×S3×D5D6.D10C4×S3×D5
kernelC4×C15⋊Q8C30.Q8Dic155C4C6.Dic10C12×Dic5Dic3×C20C4×Dic15C2×C15⋊Q8C15⋊Q8C4×Dic5C60C4×Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12Dic5C20Dic3C12C10C6C2×C4C4C22C2C2
# reps1111111181222124244884824244

Matrix representation of C4×C15⋊Q8 in GL5(𝔽61)

500000
011000
001100
00010
00001
,
10000
0606000
01000
000018
0004417
,
600000
0215900
0384000
00010
00001
,
10000
0381500
0462300
0001452
0004247

G:=sub<GL(5,GF(61))| [50,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,1,0,0,0,60,0,0,0,0,0,0,0,44,0,0,0,18,17],[60,0,0,0,0,0,21,38,0,0,0,59,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,38,46,0,0,0,15,23,0,0,0,0,0,14,42,0,0,0,52,47] >;

C4×C15⋊Q8 in GAP, Magma, Sage, TeX

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

G:=Group("C4xC15:Q8");
// GroupNames label

G:=SmallGroup(480,543);
// by ID

G=gap.SmallGroup(480,543);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,120,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^15=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^11,d*b*d^-1=b^4,d*c*d^-1=c^-1>;
// generators/relations

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