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G = Dic15⋊6Q8  order 480 = 25·3·5

4th semidirect product of Dic15 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic15⋊6Q8, Dic10⋊8Dic3, C15⋊7(C4×Q8), C5⋊3(Q8×Dic3), C6.3(Q8×D5), C30.8(C2×Q8), C10.3(S3×Q8), C12.24(C4×D5), C60.132(C2×C4), C30.3(C4○D4), (C2×C20).105D6, C2.2(D15⋊Q8), C4⋊Dic3.11D5, C4.10(D5×Dic3), (C3×Dic10)⋊12C4, (C2×C12).107D10, C6.2(D4⋊2D5), C3⋊5(Dic5⋊3Q8), (C2×C30).21C23, (C2×Dic5).85D6, (C6×Dic10).8C2, (C2×Dic10).9S3, C20.33(C2×Dic3), C30.Q8.5C2, C2.1(D12⋊D5), C30.112(C22×C4), (C2×C60).191C22, (Dic3×Dic5).8C2, (C4×Dic15).12C2, (C2×Dic3).76D10, Dic5.6(C2×Dic3), (C6×Dic5).8C22, C10.20(Q8⋊3S3), (C10×Dic3).7C22, C10.20(C22×Dic3), (C2×Dic15).180C22, C6.83(C2×C4×D5), C2.9(C2×D5×Dic3), C22.25(C2×S3×D5), (C2×C4).201(S3×D5), (C5×C4⋊Dic3).10C2, (C2×C6).33(C22×D5), (C2×C10).33(C22×S3), (C3×Dic5).10(C2×C4), SmallGroup(480,407)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — Dic15⋊6Q8
Chief series
C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic15⋊6Q8
Lower central
C15 — C30 — Dic15⋊6Q8
Upper central
C1 — C22 — C2×C4

Generators and relations for Dic15⋊6Q8
 G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=a-1, ac=ca, dad-1=a19, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 508 in 140 conjugacy classes, 70 normal (32 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C30, C4×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, C4⋊Dic3, C4⋊Dic3, C6×Q8, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C5×C4⋊C4, C2×Dic10, Q8×Dic3, C3×Dic10, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, Dic5⋊3Q8, Dic3×Dic5, C30.Q8, C5×C4⋊Dic3, C4×Dic15, C6×Dic10, Dic15⋊6Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D5, Dic3, D6, C22×C4, C2×Q8, C4○D4, D10, C2×Dic3, C22×S3, C4×Q8, C4×D5, C22×D5, S3×Q8, Q8⋊3S3, C22×Dic3, S3×D5, C2×C4×D5, D4⋊2D5, Q8×D5, Q8×Dic3, D5×Dic3, C2×S3×D5, Dic5⋊3Q8, D12⋊D5, D15⋊Q8, C2×D5×Dic3, Dic15⋊6Q8

Smallest permutation representation of Dic15⋊6Q8
►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)

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

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

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

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444444444445566610···1012121212121215152020202020···2030···3060···60
size1111222666610101010151515153030222222···2442020202044444412···124···44···4

66 irreducible representations

dim11111112222222222444444444
type+++++++-+-++++-++---+
imageC1C2C2C2C2C2C4S3Q8D5Dic3D6D6C4○D4D10D10C4×D5S3×Q8Q8⋊3S3S3×D5D4⋊2D5Q8×D5D5×Dic3C2×S3×D5D12⋊D5D15⋊Q8
kernelDic15⋊6Q8Dic3×Dic5C30.Q8C5×C4⋊Dic3C4×Dic15C6×Dic10C3×Dic10C2×Dic10Dic15C4⋊Dic3Dic10C2×Dic5C2×C20C30C2×Dic3C2×C12C12C10C10C2×C4C6C6C4C22C2C2
# reps12211181224212428112224244

Matrix representation of Dic15⋊6Q8 ►in GL7(𝔽61)

60000000
0100000
0010000
00044100
000166000
00000601
00000600
,
50000000
0100000
0010000
000552200
00040600
000003451
000002427
,
1000000
0010000
06000000
0001000
0000100
0000010
0000001
,
1000000
022530000
053390000
000552200
00040600
0000010
0000001

G:=sub<GL(7,GF(61))| [60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,44,16,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,60,0,0,0,0,0,1,0],[50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,55,40,0,0,0,0,0,22,6,0,0,0,0,0,0,0,34,24,0,0,0,0,0,51,27],[1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,22,53,0,0,0,0,0,53,39,0,0,0,0,0,0,0,55,40,0,0,0,0,0,22,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;
 

Dic15⋊6Q8 in GAP, Magma, Sage, TeX 

{\rm Dic}_{15}\rtimes_6Q_8
 
% in TeX
 
G:=Group("Dic15:6Q8");
 
// GroupNames label
 
G:=SmallGroup(480,407);
 
// by ID
 
G=gap.SmallGroup(480,407);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,422,219,100,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^30=c^4=1,b^2=a^15,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^19,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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