Copied to
clipboard

## G = Dic15⋊4Q8order 480 = 25·3·5

### 2nd semidirect product of Dic15 and 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 — C2×Dic15 — C4×Dic15 — Dic15⋊4Q8
 Lower central C15 — C2×C30 — Dic15⋊4Q8
 Upper central C1 — C22 — C2×Q8

Generators and relations for Dic154Q8
G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 628 in 136 conjugacy classes, 59 normal (23 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×8], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10, C10 [×2], Dic3 [×6], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8, C2×Q8, Dic5 [×6], C20 [×2], C20 [×2], C2×C10, Dic6 [×2], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C3×Q8 [×2], C30, C30 [×2], C4⋊Q8, Dic10 [×2], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C5×Q8 [×2], C4×Dic3, Dic3⋊C4 [×4], C2×Dic6, C6×Q8, Dic15 [×4], Dic15 [×2], C60 [×2], C60 [×2], C2×C30, C4×Dic5, C10.D4 [×4], C2×Dic10, Q8×C10, Dic3⋊Q8, Dic30 [×2], C2×Dic15 [×4], C2×C60, C2×C60 [×2], Q8×C15 [×2], Dic5⋊Q8, C4×Dic15, C30.4Q8 [×4], C2×Dic30, Q8×C30, Dic154Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D5, D6 [×3], C2×D4, C2×Q8 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, D15, C4⋊Q8, C5⋊D4 [×2], C22×D5, S3×Q8 [×2], C2×C3⋊D4, D30 [×3], Q8×D5 [×2], C2×C5⋊D4, Dic3⋊Q8, C157D4 [×2], C22×D15, Dic5⋊Q8, Q8×D15 [×2], C2×C157D4, Dic154Q8

Smallest permutation representation of Dic154Q8
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)
(1 143 16 128)(2 142 17 127)(3 141 18 126)(4 140 19 125)(5 139 20 124)(6 138 21 123)(7 137 22 122)(8 136 23 121)(9 135 24 150)(10 134 25 149)(11 133 26 148)(12 132 27 147)(13 131 28 146)(14 130 29 145)(15 129 30 144)(31 415 46 400)(32 414 47 399)(33 413 48 398)(34 412 49 397)(35 411 50 396)(36 410 51 395)(37 409 52 394)(38 408 53 393)(39 407 54 392)(40 406 55 391)(41 405 56 420)(42 404 57 419)(43 403 58 418)(44 402 59 417)(45 401 60 416)(61 291 76 276)(62 290 77 275)(63 289 78 274)(64 288 79 273)(65 287 80 272)(66 286 81 271)(67 285 82 300)(68 284 83 299)(69 283 84 298)(70 282 85 297)(71 281 86 296)(72 280 87 295)(73 279 88 294)(74 278 89 293)(75 277 90 292)(91 257 106 242)(92 256 107 241)(93 255 108 270)(94 254 109 269)(95 253 110 268)(96 252 111 267)(97 251 112 266)(98 250 113 265)(99 249 114 264)(100 248 115 263)(101 247 116 262)(102 246 117 261)(103 245 118 260)(104 244 119 259)(105 243 120 258)(151 199 166 184)(152 198 167 183)(153 197 168 182)(154 196 169 181)(155 195 170 210)(156 194 171 209)(157 193 172 208)(158 192 173 207)(159 191 174 206)(160 190 175 205)(161 189 176 204)(162 188 177 203)(163 187 178 202)(164 186 179 201)(165 185 180 200)(211 443 226 428)(212 442 227 427)(213 441 228 426)(214 440 229 425)(215 439 230 424)(216 438 231 423)(217 437 232 422)(218 436 233 421)(219 435 234 450)(220 434 235 449)(221 433 236 448)(222 432 237 447)(223 431 238 446)(224 430 239 445)(225 429 240 444)(301 364 316 379)(302 363 317 378)(303 362 318 377)(304 361 319 376)(305 390 320 375)(306 389 321 374)(307 388 322 373)(308 387 323 372)(309 386 324 371)(310 385 325 370)(311 384 326 369)(312 383 327 368)(313 382 328 367)(314 381 329 366)(315 380 330 365)(331 455 346 470)(332 454 347 469)(333 453 348 468)(334 452 349 467)(335 451 350 466)(336 480 351 465)(337 479 352 464)(338 478 353 463)(339 477 354 462)(340 476 355 461)(341 475 356 460)(342 474 357 459)(343 473 358 458)(344 472 359 457)(345 471 360 456)
(1 183 378 270)(2 184 379 241)(3 185 380 242)(4 186 381 243)(5 187 382 244)(6 188 383 245)(7 189 384 246)(8 190 385 247)(9 191 386 248)(10 192 387 249)(11 193 388 250)(12 194 389 251)(13 195 390 252)(14 196 361 253)(15 197 362 254)(16 198 363 255)(17 199 364 256)(18 200 365 257)(19 201 366 258)(20 202 367 259)(21 203 368 260)(22 204 369 261)(23 205 370 262)(24 206 371 263)(25 207 372 264)(26 208 373 265)(27 209 374 266)(28 210 375 267)(29 181 376 268)(30 182 377 269)(31 279 223 467)(32 280 224 468)(33 281 225 469)(34 282 226 470)(35 283 227 471)(36 284 228 472)(37 285 229 473)(38 286 230 474)(39 287 231 475)(40 288 232 476)(41 289 233 477)(42 290 234 478)(43 291 235 479)(44 292 236 480)(45 293 237 451)(46 294 238 452)(47 295 239 453)(48 296 240 454)(49 297 211 455)(50 298 212 456)(51 299 213 457)(52 300 214 458)(53 271 215 459)(54 272 216 460)(55 273 217 461)(56 274 218 462)(57 275 219 463)(58 276 220 464)(59 277 221 465)(60 278 222 466)(61 449 337 403)(62 450 338 404)(63 421 339 405)(64 422 340 406)(65 423 341 407)(66 424 342 408)(67 425 343 409)(68 426 344 410)(69 427 345 411)(70 428 346 412)(71 429 347 413)(72 430 348 414)(73 431 349 415)(74 432 350 416)(75 433 351 417)(76 434 352 418)(77 435 353 419)(78 436 354 420)(79 437 355 391)(80 438 356 392)(81 439 357 393)(82 440 358 394)(83 441 359 395)(84 442 360 396)(85 443 331 397)(86 444 332 398)(87 445 333 399)(88 446 334 400)(89 447 335 401)(90 448 336 402)(91 126 180 315)(92 127 151 316)(93 128 152 317)(94 129 153 318)(95 130 154 319)(96 131 155 320)(97 132 156 321)(98 133 157 322)(99 134 158 323)(100 135 159 324)(101 136 160 325)(102 137 161 326)(103 138 162 327)(104 139 163 328)(105 140 164 329)(106 141 165 330)(107 142 166 301)(108 143 167 302)(109 144 168 303)(110 145 169 304)(111 146 170 305)(112 147 171 306)(113 148 172 307)(114 149 173 308)(115 150 174 309)(116 121 175 310)(117 122 176 311)(118 123 177 312)(119 124 178 313)(120 125 179 314)
(1 404 378 450)(2 405 379 421)(3 406 380 422)(4 407 381 423)(5 408 382 424)(6 409 383 425)(7 410 384 426)(8 411 385 427)(9 412 386 428)(10 413 387 429)(11 414 388 430)(12 415 389 431)(13 416 390 432)(14 417 361 433)(15 418 362 434)(16 419 363 435)(17 420 364 436)(18 391 365 437)(19 392 366 438)(20 393 367 439)(21 394 368 440)(22 395 369 441)(23 396 370 442)(24 397 371 443)(25 398 372 444)(26 399 373 445)(27 400 374 446)(28 401 375 447)(29 402 376 448)(30 403 377 449)(31 306 223 147)(32 307 224 148)(33 308 225 149)(34 309 226 150)(35 310 227 121)(36 311 228 122)(37 312 229 123)(38 313 230 124)(39 314 231 125)(40 315 232 126)(41 316 233 127)(42 317 234 128)(43 318 235 129)(44 319 236 130)(45 320 237 131)(46 321 238 132)(47 322 239 133)(48 323 240 134)(49 324 211 135)(50 325 212 136)(51 326 213 137)(52 327 214 138)(53 328 215 139)(54 329 216 140)(55 330 217 141)(56 301 218 142)(57 302 219 143)(58 303 220 144)(59 304 221 145)(60 305 222 146)(61 182 337 269)(62 183 338 270)(63 184 339 241)(64 185 340 242)(65 186 341 243)(66 187 342 244)(67 188 343 245)(68 189 344 246)(69 190 345 247)(70 191 346 248)(71 192 347 249)(72 193 348 250)(73 194 349 251)(74 195 350 252)(75 196 351 253)(76 197 352 254)(77 198 353 255)(78 199 354 256)(79 200 355 257)(80 201 356 258)(81 202 357 259)(82 203 358 260)(83 204 359 261)(84 205 360 262)(85 206 331 263)(86 207 332 264)(87 208 333 265)(88 209 334 266)(89 210 335 267)(90 181 336 268)(91 288 180 476)(92 289 151 477)(93 290 152 478)(94 291 153 479)(95 292 154 480)(96 293 155 451)(97 294 156 452)(98 295 157 453)(99 296 158 454)(100 297 159 455)(101 298 160 456)(102 299 161 457)(103 300 162 458)(104 271 163 459)(105 272 164 460)(106 273 165 461)(107 274 166 462)(108 275 167 463)(109 276 168 464)(110 277 169 465)(111 278 170 466)(112 279 171 467)(113 280 172 468)(114 281 173 469)(115 282 174 470)(116 283 175 471)(117 284 176 472)(118 285 177 473)(119 286 178 474)(120 287 179 475)

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

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

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

84 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 ··· 12F 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X 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 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 4 4 30 30 30 30 60 60 2 2 2 2 2 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + + + + - - - image C1 C2 C2 C2 C2 S3 Q8 D4 D5 D6 D10 C3⋊D4 D15 C5⋊D4 D30 C15⋊7D4 S3×Q8 Q8×D5 Q8×D15 kernel Dic15⋊4Q8 C4×Dic15 C30.4Q8 C2×Dic30 Q8×C30 Q8×C10 Dic15 C60 C6×Q8 C2×C20 C2×C12 C20 C2×Q8 C12 C2×C4 C4 C10 C6 C2 # reps 1 1 4 1 1 1 4 2 2 3 6 4 4 8 12 16 2 4 8

Matrix representation of Dic154Q8 in GL6(𝔽61)

 33 24 0 0 0 0 37 14 0 0 0 0 0 0 58 15 0 0 0 0 52 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 53 55 0 0 0 0 41 8 0 0 0 0 0 0 28 3 0 0 0 0 23 33 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 35 0 0 0 0 40 59 0 0 0 0 0 0 1 53 0 0 0 0 46 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 30 7 0 0 0 0 2 31

G:=sub<GL(6,GF(61))| [33,37,0,0,0,0,24,14,0,0,0,0,0,0,58,52,0,0,0,0,15,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[53,41,0,0,0,0,55,8,0,0,0,0,0,0,28,23,0,0,0,0,3,33,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,40,0,0,0,0,35,59,0,0,0,0,0,0,1,46,0,0,0,0,53,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,30,2,0,0,0,0,7,31] >;

Dic154Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_4Q_8
% in TeX

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

G:=SmallGroup(480,909);
// by ID

G=gap.SmallGroup(480,909);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,422,135,58,2693,18822]);
// 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,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽