Copied to
clipboard

G = Dic15.2Q8order 480 = 25·3·5

2nd non-split extension by Dic15 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic15.2Q8, C6.5(Q8×D5), (C2×C20).5D6, C10.5(S3×Q8), C4⋊Dic5.4S3, C30.14(C2×Q8), C2.9(D15⋊Q8), Dic3⋊C4.8D5, C52(Dic3.Q8), (C2×Dic5).3D6, C154(C42.C2), C6.45(C4○D20), (C2×C12).217D10, (C2×C30).29C23, C30.Q8.8C2, Dic155C4.7C2, C30.4Q8.7C2, C30.103(C4○D4), C10.65(C4○D12), C6.34(D42D5), C2.6(D205S3), (C2×C60).248C22, (C2×Dic3).80D10, C36(Dic5.Q8), C10.18(D42S3), (Dic3×Dic5).14C2, C2.8(Dic3.D10), (C6×Dic5).14C22, (C10×Dic3).14C22, (C2×Dic15).36C22, (C2×C4).27(S3×D5), C22.121(C2×S3×D5), (C5×Dic3⋊C4).8C2, (C3×C4⋊Dic5).16C2, (C2×C6).41(C22×D5), (C2×C10).41(C22×S3), SmallGroup(480,415)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic15.2Q8
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic15.2Q8
C15C2×C30 — Dic15.2Q8
C1C22C2×C4

Generators and relations for Dic15.2Q8
 G = < a,b,c,d | a30=c4=1, b2=a15, d2=a15c2, bab-1=a-1, ac=ca, dad-1=a19, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 460 in 112 conjugacy classes, 46 normal (44 characteristic)
C1, C2 [×3], C3, C4 [×8], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], C10 [×3], Dic3 [×5], C12 [×3], C2×C6, C15, C42, C4⋊C4 [×6], Dic5 [×5], C20 [×3], C2×C10, C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C42.C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3, Dic3⋊C4, Dic3⋊C4 [×3], C4⋊Dic3, C3×C4⋊C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], Dic15, C60, C2×C30, C4×Dic5, C10.D4 [×4], C4⋊Dic5, C5×C4⋊C4, Dic3.Q8, C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, Dic5.Q8, Dic3×Dic5, C30.Q8, Dic155C4 [×2], C3×C4⋊Dic5, C5×Dic3⋊C4, C30.4Q8, Dic15.2Q8
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D5, D6 [×3], C2×Q8, C4○D4 [×2], D10 [×3], 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, D205S3, D15⋊Q8, Dic3.D10, Dic15.2Q8

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

dim11111112222222222444444444
type++++++++-+++++--+--+-
imageC1C2C2C2C2C2C2S3Q8D5D6D6C4○D4D10D10C4○D12C4○D20D42S3S3×Q8S3×D5D42D5Q8×D5C2×S3×D5D205S3D15⋊Q8Dic3.D10
kernelDic15.2Q8Dic3×Dic5C30.Q8Dic155C4C3×C4⋊Dic5C5×Dic3⋊C4C30.4Q8C4⋊Dic5Dic15Dic3⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C10C6C10C10C2×C4C6C6C22C2C2C2
# reps11121111222144248112222444

Matrix representation of Dic15.2Q8 in GL6(𝔽61)

6000000
0600000
0034000
0049900
0000060
000011
,
21300000
30400000
00322600
00242900
0000050
0000500
,
010000
6000000
0060000
0006000
00005243
0000189
,
13360000
36480000
00293500
00373200
0000500
0000050

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,34,49,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,60,1],[21,30,0,0,0,0,30,40,0,0,0,0,0,0,32,24,0,0,0,0,26,29,0,0,0,0,0,0,0,50,0,0,0,0,50,0],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,52,18,0,0,0,0,43,9],[13,36,0,0,0,0,36,48,0,0,0,0,0,0,29,37,0,0,0,0,35,32,0,0,0,0,0,0,50,0,0,0,0,0,0,50] >;

Dic15.2Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}._2Q_8
% in TeX

G:=Group("Dic15.2Q8");
// GroupNames label

G:=SmallGroup(480,415);
// by ID

G=gap.SmallGroup(480,415);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,254,219,58,1356,18822]);
// 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,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽