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G = Dic15⋊Q8order 480 = 25·3·5

2nd semidirect product of Dic15 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic152Q8, Dic52Dic6, C153(C4⋊Q8), C55(C12⋊Q8), C10.2(S3×Q8), C30.6(C2×Q8), C6.21(Q8×D5), (C3×Dic5)⋊2Q8, (C2×C12).2D10, C2.7(D5×Dic6), (C2×C20).217D6, C10.125(S3×D4), C30.101(C2×D4), C2.7(D15⋊Q8), (C5×Dic3).5D4, (C2×Dic6).3D5, C10.3(C2×Dic6), C32(Dic5⋊Q8), (C2×C30).19C23, (C2×Dic5).83D6, C10.D4.7S3, C30.Q8.3C2, Dic155C4.3C2, C30.4Q8.9C2, (C2×C60).310C22, (C10×Dic6).11C2, (C2×Dic3).75D10, (Dic3×Dic5).6C2, Dic3.1(C5⋊D4), (C6×Dic5).6C22, (C10×Dic3).6C22, (C2×Dic15).27C22, C2.8(S3×C5⋊D4), (C2×C15⋊Q8).6C2, (C2×C4).22(S3×D5), C6.26(C2×C5⋊D4), C22.116(C2×S3×D5), (C2×C6).31(C22×D5), (C2×C10).31(C22×S3), (C3×C10.D4).9C2, SmallGroup(480,405)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic15⋊Q8
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic15⋊Q8
C15C2×C30 — Dic15⋊Q8
C1C22C2×C4

Generators and relations for Dic15⋊Q8
 G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=a-1, ac=ca, dad-1=a11, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 588 in 136 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C3, C4 [×10], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×2], Dic3 [×4], C12 [×4], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×2], Dic5 [×4], C20 [×4], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4⋊Q8, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C2×Dic6, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15 [×2], Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4 [×3], C2×Dic10, Q8×C10, C12⋊Q8, C15⋊Q8 [×2], C6×Dic5 [×2], C5×Dic6 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, Dic5⋊Q8, Dic3×Dic5, C30.Q8, Dic155C4, C3×C10.D4, C30.4Q8, C2×C15⋊Q8, C10×Dic6, Dic15⋊Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D5, D6 [×3], C2×D4, C2×Q8 [×2], D10 [×3], Dic6 [×2], C22×S3, C4⋊Q8, C5⋊D4 [×2], C22×D5, C2×Dic6, S3×D4, S3×Q8, S3×D5, Q8×D5 [×2], C2×C5⋊D4, C12⋊Q8, C2×S3×D5, Dic5⋊Q8, D5×Dic6, D15⋊Q8, S3×C5⋊D4, Dic15⋊Q8

Smallest permutation representation of Dic15⋊Q8
Regular action on 480 points
Generators in S480
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330)(331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390)(391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450)(451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480)
(1 327 16 312)(2 326 17 311)(3 325 18 310)(4 324 19 309)(5 323 20 308)(6 322 21 307)(7 321 22 306)(8 320 23 305)(9 319 24 304)(10 318 25 303)(11 317 26 302)(12 316 27 301)(13 315 28 330)(14 314 29 329)(15 313 30 328)(31 440 46 425)(32 439 47 424)(33 438 48 423)(34 437 49 422)(35 436 50 421)(36 435 51 450)(37 434 52 449)(38 433 53 448)(39 432 54 447)(40 431 55 446)(41 430 56 445)(42 429 57 444)(43 428 58 443)(44 427 59 442)(45 426 60 441)(61 102 76 117)(62 101 77 116)(63 100 78 115)(64 99 79 114)(65 98 80 113)(66 97 81 112)(67 96 82 111)(68 95 83 110)(69 94 84 109)(70 93 85 108)(71 92 86 107)(72 91 87 106)(73 120 88 105)(74 119 89 104)(75 118 90 103)(121 263 136 248)(122 262 137 247)(123 261 138 246)(124 260 139 245)(125 259 140 244)(126 258 141 243)(127 257 142 242)(128 256 143 241)(129 255 144 270)(130 254 145 269)(131 253 146 268)(132 252 147 267)(133 251 148 266)(134 250 149 265)(135 249 150 264)(151 412 166 397)(152 411 167 396)(153 410 168 395)(154 409 169 394)(155 408 170 393)(156 407 171 392)(157 406 172 391)(158 405 173 420)(159 404 174 419)(160 403 175 418)(161 402 176 417)(162 401 177 416)(163 400 178 415)(164 399 179 414)(165 398 180 413)(181 231 196 216)(182 230 197 215)(183 229 198 214)(184 228 199 213)(185 227 200 212)(186 226 201 211)(187 225 202 240)(188 224 203 239)(189 223 204 238)(190 222 205 237)(191 221 206 236)(192 220 207 235)(193 219 208 234)(194 218 209 233)(195 217 210 232)(271 453 286 468)(272 452 287 467)(273 451 288 466)(274 480 289 465)(275 479 290 464)(276 478 291 463)(277 477 292 462)(278 476 293 461)(279 475 294 460)(280 474 295 459)(281 473 296 458)(282 472 297 457)(283 471 298 456)(284 470 299 455)(285 469 300 454)(331 372 346 387)(332 371 347 386)(333 370 348 385)(334 369 349 384)(335 368 350 383)(336 367 351 382)(337 366 352 381)(338 365 353 380)(339 364 354 379)(340 363 355 378)(341 362 356 377)(342 361 357 376)(343 390 358 375)(344 389 359 374)(345 388 360 373)
(1 226 42 362)(2 227 43 363)(3 228 44 364)(4 229 45 365)(5 230 46 366)(6 231 47 367)(7 232 48 368)(8 233 49 369)(9 234 50 370)(10 235 51 371)(11 236 52 372)(12 237 53 373)(13 238 54 374)(14 239 55 375)(15 240 56 376)(16 211 57 377)(17 212 58 378)(18 213 59 379)(19 214 60 380)(20 215 31 381)(21 216 32 382)(22 217 33 383)(23 218 34 384)(24 219 35 385)(25 220 36 386)(26 221 37 387)(27 222 38 388)(28 223 39 389)(29 224 40 390)(30 225 41 361)(61 259 403 461)(62 260 404 462)(63 261 405 463)(64 262 406 464)(65 263 407 465)(66 264 408 466)(67 265 409 467)(68 266 410 468)(69 267 411 469)(70 268 412 470)(71 269 413 471)(72 270 414 472)(73 241 415 473)(74 242 416 474)(75 243 417 475)(76 244 418 476)(77 245 419 477)(78 246 420 478)(79 247 391 479)(80 248 392 480)(81 249 393 451)(82 250 394 452)(83 251 395 453)(84 252 396 454)(85 253 397 455)(86 254 398 456)(87 255 399 457)(88 256 400 458)(89 257 401 459)(90 258 402 460)(91 144 164 282)(92 145 165 283)(93 146 166 284)(94 147 167 285)(95 148 168 286)(96 149 169 287)(97 150 170 288)(98 121 171 289)(99 122 172 290)(100 123 173 291)(101 124 174 292)(102 125 175 293)(103 126 176 294)(104 127 177 295)(105 128 178 296)(106 129 179 297)(107 130 180 298)(108 131 151 299)(109 132 152 300)(110 133 153 271)(111 134 154 272)(112 135 155 273)(113 136 156 274)(114 137 157 275)(115 138 158 276)(116 139 159 277)(117 140 160 278)(118 141 161 279)(119 142 162 280)(120 143 163 281)(181 424 336 322)(182 425 337 323)(183 426 338 324)(184 427 339 325)(185 428 340 326)(186 429 341 327)(187 430 342 328)(188 431 343 329)(189 432 344 330)(190 433 345 301)(191 434 346 302)(192 435 347 303)(193 436 348 304)(194 437 349 305)(195 438 350 306)(196 439 351 307)(197 440 352 308)(198 441 353 309)(199 442 354 310)(200 443 355 311)(201 444 356 312)(202 445 357 313)(203 446 358 314)(204 447 359 315)(205 448 360 316)(206 449 331 317)(207 450 332 318)(208 421 333 319)(209 422 334 320)(210 423 335 321)
(1 266 42 468)(2 247 43 479)(3 258 44 460)(4 269 45 471)(5 250 46 452)(6 261 47 463)(7 242 48 474)(8 253 49 455)(9 264 50 466)(10 245 51 477)(11 256 52 458)(12 267 53 469)(13 248 54 480)(14 259 55 461)(15 270 56 472)(16 251 57 453)(17 262 58 464)(18 243 59 475)(19 254 60 456)(20 265 31 467)(21 246 32 478)(22 257 33 459)(23 268 34 470)(24 249 35 451)(25 260 36 462)(26 241 37 473)(27 252 38 454)(28 263 39 465)(29 244 40 476)(30 255 41 457)(61 375 403 239)(62 386 404 220)(63 367 405 231)(64 378 406 212)(65 389 407 223)(66 370 408 234)(67 381 409 215)(68 362 410 226)(69 373 411 237)(70 384 412 218)(71 365 413 229)(72 376 414 240)(73 387 415 221)(74 368 416 232)(75 379 417 213)(76 390 418 224)(77 371 419 235)(78 382 420 216)(79 363 391 227)(80 374 392 238)(81 385 393 219)(82 366 394 230)(83 377 395 211)(84 388 396 222)(85 369 397 233)(86 380 398 214)(87 361 399 225)(88 372 400 236)(89 383 401 217)(90 364 402 228)(91 342 164 187)(92 353 165 198)(93 334 166 209)(94 345 167 190)(95 356 168 201)(96 337 169 182)(97 348 170 193)(98 359 171 204)(99 340 172 185)(100 351 173 196)(101 332 174 207)(102 343 175 188)(103 354 176 199)(104 335 177 210)(105 346 178 191)(106 357 179 202)(107 338 180 183)(108 349 151 194)(109 360 152 205)(110 341 153 186)(111 352 154 197)(112 333 155 208)(113 344 156 189)(114 355 157 200)(115 336 158 181)(116 347 159 192)(117 358 160 203)(118 339 161 184)(119 350 162 195)(120 331 163 206)(121 447 289 315)(122 428 290 326)(123 439 291 307)(124 450 292 318)(125 431 293 329)(126 442 294 310)(127 423 295 321)(128 434 296 302)(129 445 297 313)(130 426 298 324)(131 437 299 305)(132 448 300 316)(133 429 271 327)(134 440 272 308)(135 421 273 319)(136 432 274 330)(137 443 275 311)(138 424 276 322)(139 435 277 303)(140 446 278 314)(141 427 279 325)(142 438 280 306)(143 449 281 317)(144 430 282 328)(145 441 283 309)(146 422 284 320)(147 433 285 301)(148 444 286 312)(149 425 287 323)(150 436 288 304)

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

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

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

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444445566610···1012121212121215152020202020···2030···3060···60
size1111246610101220303060222222···2442020202044444412···124···44···4

60 irreducible representations

dim111111112222222222244444444
type++++++++++--+++++-+-+-+-
imageC1C2C2C2C2C2C2C2S3D4Q8Q8D5D6D6D10D10Dic6C5⋊D4S3×D4S3×Q8S3×D5Q8×D5C2×S3×D5D5×Dic6D15⋊Q8S3×C5⋊D4
kernelDic15⋊Q8Dic3×Dic5C30.Q8Dic155C4C3×C10.D4C30.4Q8C2×C15⋊Q8C10×Dic6C10.D4C5×Dic3C3×Dic5Dic15C2×Dic6C2×Dic5C2×C20C2×Dic3C2×C12Dic5Dic3C10C10C2×C4C6C22C2C2C2
# reps111111111222221424811242444

Matrix representation of Dic15⋊Q8 in GL6(𝔽61)

010000
6010000
001000
000100
0000431
0000600
,
53490000
4180000
001000
000100
0000168
00005245
,
23150000
46380000
008300
00195300
000010
000001
,
53490000
4180000
00363700
00212500
0000600
0000060

G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,60,0,0,0,0,1,0],[53,41,0,0,0,0,49,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,52,0,0,0,0,8,45],[23,46,0,0,0,0,15,38,0,0,0,0,0,0,8,19,0,0,0,0,3,53,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[53,41,0,0,0,0,49,8,0,0,0,0,0,0,36,21,0,0,0,0,37,25,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

Dic15⋊Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes Q_8
% in TeX

G:=Group("Dic15:Q8");
// GroupNames label

G:=SmallGroup(480,405);
// by ID

G=gap.SmallGroup(480,405);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,422,135,58,1356,18822]);
// 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^11,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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