Copied to
clipboard

## G = Dic15.4Q8order 480 = 25·3·5

### 2nd non-split extension by Dic15 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic15.4Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic15⋊5C4 — Dic15.4Q8
 Lower central C15 — C2×C30 — Dic15.4Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic15.4Q8
G = < a,b,c,d | a30=c4=1, b2=a15, d2=a15c2, bab-1=a-1, ac=ca, dad-1=a19, bc=cb, dbd-1=a15b, dcd-1=a15c-1 >

Subgroups: 460 in 112 conjugacy classes, 46 normal (44 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C2×C4, C10, Dic3, C12, C2×C6, C15, C42, C4⋊C4, Dic5, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C5×C4⋊C4, Dic3.Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, Dic5.Q8, C30.Q8, Dic155C4, C6.Dic10, C3×C10.D4, C5×Dic3⋊C4, C4×Dic15, Dic15.4Q8
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, C22×S3, C42.C2, C22×D5, C4○D12, D42S3, S3×Q8, S3×D5, C4○D20, D42D5, Q8×D5, Dic3.Q8, C2×S3×D5, Dic5.Q8, D15⋊Q8, D6.D10, C30.C23, Dic15.4Q8

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

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

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

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

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 2 2 12 12 20 20 30 30 30 30 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 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + - - + - - + - image C1 C2 C2 C2 C2 C2 C2 S3 Q8 D5 D6 D6 C4○D4 D10 D10 C4○D12 C4○D20 D4⋊2S3 S3×Q8 S3×D5 D4⋊2D5 Q8×D5 C2×S3×D5 D15⋊Q8 D6.D10 C30.C23 kernel Dic15.4Q8 C30.Q8 Dic15⋊5C4 C6.Dic10 C3×C10.D4 C5×Dic3⋊C4 C4×Dic15 C10.D4 Dic15 Dic3⋊C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C10 C6 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 2 1 1 1 1 1 2 2 2 1 4 4 2 4 8 1 1 2 2 2 2 4 4 4

Matrix representation of Dic15.4Q8 in GL6(𝔽61)

 0 60 0 0 0 0 1 44 0 0 0 0 0 0 1 5 0 0 0 0 36 59 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 53 51 0 0 0 0 37 8 0 0 0 0 0 0 43 33 0 0 0 0 5 18 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 50 0 0 0 0 0 0 50 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 10 0 0 0 0 12 1
,
 28 37 0 0 0 0 25 33 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 21 54 0 0 0 0 37 40

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,44,0,0,0,0,0,0,1,36,0,0,0,0,5,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[53,37,0,0,0,0,51,8,0,0,0,0,0,0,43,5,0,0,0,0,33,18,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,12,0,0,0,0,10,1],[28,25,0,0,0,0,37,33,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,21,37,0,0,0,0,54,40] >;

Dic15.4Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}._4Q_8
% in TeX

G:=Group("Dic15.4Q8");
// GroupNames label

G:=SmallGroup(480,458);
// by ID

G=gap.SmallGroup(480,458);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,422,219,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=1,b^2=a^15,d^2=a^15*c^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^19,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^-1>;
// generators/relations

׿
×
𝔽