Copied to
clipboard

## G = Dic15⋊5Q8order 480 = 25·3·5

### 3rd semidirect product of Dic15 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic15⋊5Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic15⋊5Q8
 Lower central C15 — C30 — Dic15⋊5Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic155Q8
G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=cac-1=a-1, dad-1=a11, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 556 in 140 conjugacy classes, 60 normal (44 characteristic)
C1, C2 [×3], C3, C4 [×11], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×2], Dic3 [×5], C12 [×4], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×2], Dic5 [×5], C20 [×4], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4×Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3 [×3], Dic3⋊C4, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15 [×4], C60, C2×C30, C4×Dic5 [×3], C10.D4, C10.D4, C5×C4⋊C4, C2×Dic10, Dic6⋊C4, C15⋊Q8 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, Dic53Q8, Dic3×Dic5 [×2], Dic155C4, C3×C10.D4, C5×Dic3⋊C4, C4×Dic15, C2×C15⋊Q8, Dic155Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D5, D6 [×3], C22×C4, C2×Q8, C4○D4, D10 [×3], C4×S3 [×2], C22×S3, C4×Q8, C4×D5 [×2], C22×D5, S3×C2×C4, D42S3, S3×Q8, S3×D5, C2×C4×D5, D42D5, Q8×D5, Dic6⋊C4, C2×S3×D5, Dic53Q8, D15⋊Q8, C4×S3×D5, C30.C23, Dic155Q8

Smallest permutation representation of Dic155Q8
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)
(1 248 16 263)(2 247 17 262)(3 246 18 261)(4 245 19 260)(5 244 20 259)(6 243 21 258)(7 242 22 257)(8 241 23 256)(9 270 24 255)(10 269 25 254)(11 268 26 253)(12 267 27 252)(13 266 28 251)(14 265 29 250)(15 264 30 249)(31 99 46 114)(32 98 47 113)(33 97 48 112)(34 96 49 111)(35 95 50 110)(36 94 51 109)(37 93 52 108)(38 92 53 107)(39 91 54 106)(40 120 55 105)(41 119 56 104)(42 118 57 103)(43 117 58 102)(44 116 59 101)(45 115 60 100)(61 215 76 230)(62 214 77 229)(63 213 78 228)(64 212 79 227)(65 211 80 226)(66 240 81 225)(67 239 82 224)(68 238 83 223)(69 237 84 222)(70 236 85 221)(71 235 86 220)(72 234 87 219)(73 233 88 218)(74 232 89 217)(75 231 90 216)(121 396 136 411)(122 395 137 410)(123 394 138 409)(124 393 139 408)(125 392 140 407)(126 391 141 406)(127 420 142 405)(128 419 143 404)(129 418 144 403)(130 417 145 402)(131 416 146 401)(132 415 147 400)(133 414 148 399)(134 413 149 398)(135 412 150 397)(151 347 166 332)(152 346 167 331)(153 345 168 360)(154 344 169 359)(155 343 170 358)(156 342 171 357)(157 341 172 356)(158 340 173 355)(159 339 174 354)(160 338 175 353)(161 337 176 352)(162 336 177 351)(163 335 178 350)(164 334 179 349)(165 333 180 348)(181 328 196 313)(182 327 197 312)(183 326 198 311)(184 325 199 310)(185 324 200 309)(186 323 201 308)(187 322 202 307)(188 321 203 306)(189 320 204 305)(190 319 205 304)(191 318 206 303)(192 317 207 302)(193 316 208 301)(194 315 209 330)(195 314 210 329)(271 447 286 432)(272 446 287 431)(273 445 288 430)(274 444 289 429)(275 443 290 428)(276 442 291 427)(277 441 292 426)(278 440 293 425)(279 439 294 424)(280 438 295 423)(281 437 296 422)(282 436 297 421)(283 435 298 450)(284 434 299 449)(285 433 300 448)(361 452 376 467)(362 451 377 466)(363 480 378 465)(364 479 379 464)(365 478 380 463)(366 477 381 462)(367 476 382 461)(368 475 383 460)(369 474 384 459)(370 473 385 458)(371 472 386 457)(372 471 387 456)(373 470 388 455)(374 469 389 454)(375 468 390 453)
(1 274 88 45)(2 273 89 44)(3 272 90 43)(4 271 61 42)(5 300 62 41)(6 299 63 40)(7 298 64 39)(8 297 65 38)(9 296 66 37)(10 295 67 36)(11 294 68 35)(12 293 69 34)(13 292 70 33)(14 291 71 32)(15 290 72 31)(16 289 73 60)(17 288 74 59)(18 287 75 58)(19 286 76 57)(20 285 77 56)(21 284 78 55)(22 283 79 54)(23 282 80 53)(24 281 81 52)(25 280 82 51)(26 279 83 50)(27 278 84 49)(28 277 85 48)(29 276 86 47)(30 275 87 46)(91 242 450 212)(92 241 421 211)(93 270 422 240)(94 269 423 239)(95 268 424 238)(96 267 425 237)(97 266 426 236)(98 265 427 235)(99 264 428 234)(100 263 429 233)(101 262 430 232)(102 261 431 231)(103 260 432 230)(104 259 433 229)(105 258 434 228)(106 257 435 227)(107 256 436 226)(108 255 437 225)(109 254 438 224)(110 253 439 223)(111 252 440 222)(112 251 441 221)(113 250 442 220)(114 249 443 219)(115 248 444 218)(116 247 445 217)(117 246 446 216)(118 245 447 215)(119 244 448 214)(120 243 449 213)(121 338 387 313)(122 337 388 312)(123 336 389 311)(124 335 390 310)(125 334 361 309)(126 333 362 308)(127 332 363 307)(128 331 364 306)(129 360 365 305)(130 359 366 304)(131 358 367 303)(132 357 368 302)(133 356 369 301)(134 355 370 330)(135 354 371 329)(136 353 372 328)(137 352 373 327)(138 351 374 326)(139 350 375 325)(140 349 376 324)(141 348 377 323)(142 347 378 322)(143 346 379 321)(144 345 380 320)(145 344 381 319)(146 343 382 318)(147 342 383 317)(148 341 384 316)(149 340 385 315)(150 339 386 314)(151 480 187 420)(152 479 188 419)(153 478 189 418)(154 477 190 417)(155 476 191 416)(156 475 192 415)(157 474 193 414)(158 473 194 413)(159 472 195 412)(160 471 196 411)(161 470 197 410)(162 469 198 409)(163 468 199 408)(164 467 200 407)(165 466 201 406)(166 465 202 405)(167 464 203 404)(168 463 204 403)(169 462 205 402)(170 461 206 401)(171 460 207 400)(172 459 208 399)(173 458 209 398)(174 457 210 397)(175 456 181 396)(176 455 182 395)(177 454 183 394)(178 453 184 393)(179 452 185 392)(180 451 186 391)
(1 369 88 133)(2 380 89 144)(3 361 90 125)(4 372 61 136)(5 383 62 147)(6 364 63 128)(7 375 64 139)(8 386 65 150)(9 367 66 131)(10 378 67 142)(11 389 68 123)(12 370 69 134)(13 381 70 145)(14 362 71 126)(15 373 72 137)(16 384 73 148)(17 365 74 129)(18 376 75 140)(19 387 76 121)(20 368 77 132)(21 379 78 143)(22 390 79 124)(23 371 80 135)(24 382 81 146)(25 363 82 127)(26 374 83 138)(27 385 84 149)(28 366 85 130)(29 377 86 141)(30 388 87 122)(31 327 290 352)(32 308 291 333)(33 319 292 344)(34 330 293 355)(35 311 294 336)(36 322 295 347)(37 303 296 358)(38 314 297 339)(39 325 298 350)(40 306 299 331)(41 317 300 342)(42 328 271 353)(43 309 272 334)(44 320 273 345)(45 301 274 356)(46 312 275 337)(47 323 276 348)(48 304 277 359)(49 315 278 340)(50 326 279 351)(51 307 280 332)(52 318 281 343)(53 329 282 354)(54 310 283 335)(55 321 284 346)(56 302 285 357)(57 313 286 338)(58 324 287 349)(59 305 288 360)(60 316 289 341)(91 199 450 163)(92 210 421 174)(93 191 422 155)(94 202 423 166)(95 183 424 177)(96 194 425 158)(97 205 426 169)(98 186 427 180)(99 197 428 161)(100 208 429 172)(101 189 430 153)(102 200 431 164)(103 181 432 175)(104 192 433 156)(105 203 434 167)(106 184 435 178)(107 195 436 159)(108 206 437 170)(109 187 438 151)(110 198 439 162)(111 209 440 173)(112 190 441 154)(113 201 442 165)(114 182 443 176)(115 193 444 157)(116 204 445 168)(117 185 446 179)(118 196 447 160)(119 207 448 171)(120 188 449 152)(211 397 241 457)(212 408 242 468)(213 419 243 479)(214 400 244 460)(215 411 245 471)(216 392 246 452)(217 403 247 463)(218 414 248 474)(219 395 249 455)(220 406 250 466)(221 417 251 477)(222 398 252 458)(223 409 253 469)(224 420 254 480)(225 401 255 461)(226 412 256 472)(227 393 257 453)(228 404 258 464)(229 415 259 475)(230 396 260 456)(231 407 261 467)(232 418 262 478)(233 399 263 459)(234 410 264 470)(235 391 265 451)(236 402 266 462)(237 413 267 473)(238 394 268 454)(239 405 269 465)(240 416 270 476)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 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 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 6 6 6 6 10 10 10 10 15 15 15 15 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

66 irreducible representations

 dim 1 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 C4 S3 Q8 D5 D6 D6 C4○D4 D10 D10 C4×S3 C4×D5 D4⋊2S3 S3×Q8 S3×D5 D4⋊2D5 Q8×D5 C2×S3×D5 D15⋊Q8 C4×S3×D5 C30.C23 kernel Dic15⋊5Q8 Dic3×Dic5 Dic15⋊5C4 C3×C10.D4 C5×Dic3⋊C4 C4×Dic15 C2×C15⋊Q8 C15⋊Q8 C10.D4 Dic15 Dic3⋊C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 Dic5 Dic3 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 2 1 1 1 1 1 8 1 2 2 2 1 2 4 2 4 8 1 1 2 2 2 2 4 4 4

Matrix representation of Dic155Q8 in GL6(𝔽61)

 44 1 0 0 0 0 16 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 60 0 0 0 0 1 0
,
 47 56 0 0 0 0 39 14 0 0 0 0 0 0 50 0 0 0 0 0 0 50 0 0 0 0 0 0 50 11 0 0 0 0 0 11
,
 14 5 0 0 0 0 22 47 0 0 0 0 0 0 1 39 0 0 0 0 50 60 0 0 0 0 0 0 60 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 19 0 0 0 0 32 1 0 0 0 0 0 0 1 60 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [44,16,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[47,39,0,0,0,0,56,14,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,11,11],[14,22,0,0,0,0,5,47,0,0,0,0,0,0,1,50,0,0,0,0,39,60,0,0,0,0,0,0,60,0,0,0,0,0,1,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,32,0,0,0,0,19,1,0,0,0,0,0,0,1,0,0,0,0,0,60,60] >;

Dic155Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_5Q_8
% in TeX

G:=Group("Dic15:5Q8");
// GroupNames label

G:=SmallGroup(480,401);
// by ID

G=gap.SmallGroup(480,401);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,254,135,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=1,b^2=a^15,d^2=c^2,b*a*b^-1=c*a*c^-1=a^-1,d*a*d^-1=a^11,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽