Copied to
clipboard

## G = Dic15⋊Q8order 480 = 25·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic15⋊Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic15⋊Q8
 Lower central C15 — C2×C30 — Dic15⋊Q8
 Upper central C1 — C22 — C2×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, C3, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C2×Dic6, C5×Dic3, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4, C2×Dic10, Q8×C10, C12⋊Q8, C15⋊Q8, C6×Dic5, C5×Dic6, C10×Dic3, C2×Dic15, C2×C60, Dic5⋊Q8, Dic3×Dic5, C30.Q8, Dic155C4, C3×C10.D4, C30.4Q8, C2×C15⋊Q8, C10×Dic6, Dic15⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, D10, Dic6, C22×S3, C4⋊Q8, C5⋊D4, C22×D5, C2×Dic6, S3×D4, S3×Q8, S3×D5, Q8×D5, 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)
(1 212 16 227)(2 211 17 226)(3 240 18 225)(4 239 19 224)(5 238 20 223)(6 237 21 222)(7 236 22 221)(8 235 23 220)(9 234 24 219)(10 233 25 218)(11 232 26 217)(12 231 27 216)(13 230 28 215)(14 229 29 214)(15 228 30 213)(31 207 46 192)(32 206 47 191)(33 205 48 190)(34 204 49 189)(35 203 50 188)(36 202 51 187)(37 201 52 186)(38 200 53 185)(39 199 54 184)(40 198 55 183)(41 197 56 182)(42 196 57 181)(43 195 58 210)(44 194 59 209)(45 193 60 208)(61 432 76 447)(62 431 77 446)(63 430 78 445)(64 429 79 444)(65 428 80 443)(66 427 81 442)(67 426 82 441)(68 425 83 440)(69 424 84 439)(70 423 85 438)(71 422 86 437)(72 421 87 436)(73 450 88 435)(74 449 89 434)(75 448 90 433)(91 326 106 311)(92 325 107 310)(93 324 108 309)(94 323 109 308)(95 322 110 307)(96 321 111 306)(97 320 112 305)(98 319 113 304)(99 318 114 303)(100 317 115 302)(101 316 116 301)(102 315 117 330)(103 314 118 329)(104 313 119 328)(105 312 120 327)(121 356 136 341)(122 355 137 340)(123 354 138 339)(124 353 139 338)(125 352 140 337)(126 351 141 336)(127 350 142 335)(128 349 143 334)(129 348 144 333)(130 347 145 332)(131 346 146 331)(132 345 147 360)(133 344 148 359)(134 343 149 358)(135 342 150 357)(151 254 166 269)(152 253 167 268)(153 252 168 267)(154 251 169 266)(155 250 170 265)(156 249 171 264)(157 248 172 263)(158 247 173 262)(159 246 174 261)(160 245 175 260)(161 244 176 259)(162 243 177 258)(163 242 178 257)(164 241 179 256)(165 270 180 255)(271 419 286 404)(272 418 287 403)(273 417 288 402)(274 416 289 401)(275 415 290 400)(276 414 291 399)(277 413 292 398)(278 412 293 397)(279 411 294 396)(280 410 295 395)(281 409 296 394)(282 408 297 393)(283 407 298 392)(284 406 299 391)(285 405 300 420)(361 476 376 461)(362 475 377 460)(363 474 378 459)(364 473 379 458)(365 472 380 457)(366 471 381 456)(367 470 382 455)(368 469 383 454)(369 468 384 453)(370 467 385 452)(371 466 386 451)(372 465 387 480)(373 464 388 479)(374 463 389 478)(375 462 390 477)
(1 319 348 63)(2 320 349 64)(3 321 350 65)(4 322 351 66)(5 323 352 67)(6 324 353 68)(7 325 354 69)(8 326 355 70)(9 327 356 71)(10 328 357 72)(11 329 358 73)(12 330 359 74)(13 301 360 75)(14 302 331 76)(15 303 332 77)(16 304 333 78)(17 305 334 79)(18 306 335 80)(19 307 336 81)(20 308 337 82)(21 309 338 83)(22 310 339 84)(23 311 340 85)(24 312 341 86)(25 313 342 87)(26 314 343 88)(27 315 344 89)(28 316 345 90)(29 317 346 61)(30 318 347 62)(31 480 162 292)(32 451 163 293)(33 452 164 294)(34 453 165 295)(35 454 166 296)(36 455 167 297)(37 456 168 298)(38 457 169 299)(39 458 170 300)(40 459 171 271)(41 460 172 272)(42 461 173 273)(43 462 174 274)(44 463 175 275)(45 464 176 276)(46 465 177 277)(47 466 178 278)(48 467 179 279)(49 468 180 280)(50 469 151 281)(51 470 152 282)(52 471 153 283)(53 472 154 284)(54 473 155 285)(55 474 156 286)(56 475 157 287)(57 476 158 288)(58 477 159 289)(59 478 160 290)(60 479 161 291)(91 137 438 235)(92 138 439 236)(93 139 440 237)(94 140 441 238)(95 141 442 239)(96 142 443 240)(97 143 444 211)(98 144 445 212)(99 145 446 213)(100 146 447 214)(101 147 448 215)(102 148 449 216)(103 149 450 217)(104 150 421 218)(105 121 422 219)(106 122 423 220)(107 123 424 221)(108 124 425 222)(109 125 426 223)(110 126 427 224)(111 127 428 225)(112 128 429 226)(113 129 430 227)(114 130 431 228)(115 131 432 229)(116 132 433 230)(117 133 434 231)(118 134 435 232)(119 135 436 233)(120 136 437 234)(181 361 247 417)(182 362 248 418)(183 363 249 419)(184 364 250 420)(185 365 251 391)(186 366 252 392)(187 367 253 393)(188 368 254 394)(189 369 255 395)(190 370 256 396)(191 371 257 397)(192 372 258 398)(193 373 259 399)(194 374 260 400)(195 375 261 401)(196 376 262 402)(197 377 263 403)(198 378 264 404)(199 379 265 405)(200 380 266 406)(201 381 267 407)(202 382 268 408)(203 383 269 409)(204 384 270 410)(205 385 241 411)(206 386 242 412)(207 387 243 413)(208 388 244 414)(209 389 245 415)(210 390 246 416)
(1 254 348 188)(2 265 349 199)(3 246 350 210)(4 257 351 191)(5 268 352 202)(6 249 353 183)(7 260 354 194)(8 241 355 205)(9 252 356 186)(10 263 357 197)(11 244 358 208)(12 255 359 189)(13 266 360 200)(14 247 331 181)(15 258 332 192)(16 269 333 203)(17 250 334 184)(18 261 335 195)(19 242 336 206)(20 253 337 187)(21 264 338 198)(22 245 339 209)(23 256 340 190)(24 267 341 201)(25 248 342 182)(26 259 343 193)(27 270 344 204)(28 251 345 185)(29 262 346 196)(30 243 347 207)(31 228 162 130)(32 239 163 141)(33 220 164 122)(34 231 165 133)(35 212 166 144)(36 223 167 125)(37 234 168 136)(38 215 169 147)(39 226 170 128)(40 237 171 139)(41 218 172 150)(42 229 173 131)(43 240 174 142)(44 221 175 123)(45 232 176 134)(46 213 177 145)(47 224 178 126)(48 235 179 137)(49 216 180 148)(50 227 151 129)(51 238 152 140)(52 219 153 121)(53 230 154 132)(54 211 155 143)(55 222 156 124)(56 233 157 135)(57 214 158 146)(58 225 159 127)(59 236 160 138)(60 217 161 149)(61 402 317 376)(62 413 318 387)(63 394 319 368)(64 405 320 379)(65 416 321 390)(66 397 322 371)(67 408 323 382)(68 419 324 363)(69 400 325 374)(70 411 326 385)(71 392 327 366)(72 403 328 377)(73 414 329 388)(74 395 330 369)(75 406 301 380)(76 417 302 361)(77 398 303 372)(78 409 304 383)(79 420 305 364)(80 401 306 375)(81 412 307 386)(82 393 308 367)(83 404 309 378)(84 415 310 389)(85 396 311 370)(86 407 312 381)(87 418 313 362)(88 399 314 373)(89 410 315 384)(90 391 316 365)(91 467 438 279)(92 478 439 290)(93 459 440 271)(94 470 441 282)(95 451 442 293)(96 462 443 274)(97 473 444 285)(98 454 445 296)(99 465 446 277)(100 476 447 288)(101 457 448 299)(102 468 449 280)(103 479 450 291)(104 460 421 272)(105 471 422 283)(106 452 423 294)(107 463 424 275)(108 474 425 286)(109 455 426 297)(110 466 427 278)(111 477 428 289)(112 458 429 300)(113 469 430 281)(114 480 431 292)(115 461 432 273)(116 472 433 284)(117 453 434 295)(118 464 435 276)(119 475 436 287)(120 456 437 298)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 4 6 6 10 10 12 20 30 30 60 2 2 2 2 2 2 ··· 2 4 4 20 20 20 20 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + - - + + + + + - + - + - + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 Q8 Q8 D5 D6 D6 D10 D10 Dic6 C5⋊D4 S3×D4 S3×Q8 S3×D5 Q8×D5 C2×S3×D5 D5×Dic6 D15⋊Q8 S3×C5⋊D4 kernel Dic15⋊Q8 Dic3×Dic5 C30.Q8 Dic15⋊5C4 C3×C10.D4 C30.4Q8 C2×C15⋊Q8 C10×Dic6 C10.D4 C5×Dic3 C3×Dic5 Dic15 C2×Dic6 C2×Dic5 C2×C20 C2×Dic3 C2×C12 Dic5 Dic3 C10 C10 C2×C4 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 4 2 4 8 1 1 2 4 2 4 4 4

Matrix representation of Dic15⋊Q8 in GL6(𝔽61)

 0 1 0 0 0 0 60 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 43 1 0 0 0 0 60 0
,
 53 49 0 0 0 0 41 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 8 0 0 0 0 52 45
,
 23 15 0 0 0 0 46 38 0 0 0 0 0 0 8 3 0 0 0 0 19 53 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 53 49 0 0 0 0 41 8 0 0 0 0 0 0 36 37 0 0 0 0 21 25 0 0 0 0 0 0 60 0 0 0 0 0 0 60

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

׿
×
𝔽