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G = Dic15.Q8order 480 = 25·3·5

1st non-split extension by Dic15 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic15.1Q8, C6.4(Q8×D5), C10.4(S3×Q8), (C2×C12).5D10, C4⋊Dic3.5D5, C30.13(C2×Q8), C30.7(C4○D4), (C2×C20).219D6, C2.8(D15⋊Q8), C56(Dic3.Q8), (C2×Dic5).1D6, C153(C42.C2), C6.62(C4○D20), (C2×C30).26C23, C10.D4.9S3, Dic155C4.6C2, C6.Dic10.6C2, C10.48(C4○D12), C2.6(D125D5), C6.17(D42D5), (C2×C60).312C22, (C2×Dic3).79D10, C34(Dic5.Q8), C30.4Q8.11C2, C10.37(D42S3), (Dic3×Dic5).13C2, C2.9(Dic5.D6), (C6×Dic5).11C22, (C10×Dic3).12C22, (C2×Dic15).33C22, (C2×C4).25(S3×D5), C22.119(C2×S3×D5), (C5×C4⋊Dic3).17C2, (C2×C6).38(C22×D5), (C2×C10).38(C22×S3), (C3×C10.D4).11C2, SmallGroup(480,412)

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic15.Q8
 G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=a-1, ac=ca, dad-1=a19, cbc-1=a15b, bd=db, dcd-1=a15c-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 [×4], C4⋊Dic3, C3×C4⋊C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4 [×3], C4⋊Dic5, C5×C4⋊C4, Dic3.Q8, C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, Dic5.Q8, Dic3×Dic5, Dic155C4 [×2], C6.Dic10, C3×C10.D4, C5×C4⋊Dic3, C30.4Q8, Dic15.Q8
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, D15⋊Q8, D125D5, Dic5.D6, Dic15.Q8

Smallest permutation representation of Dic15.Q8
Regular action on 480 points
Generators in S480
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)(301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330)(331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390)(391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420)(421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450)(451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480)
(1 120 16 105)(2 119 17 104)(3 118 18 103)(4 117 19 102)(5 116 20 101)(6 115 21 100)(7 114 22 99)(8 113 23 98)(9 112 24 97)(10 111 25 96)(11 110 26 95)(12 109 27 94)(13 108 28 93)(14 107 29 92)(15 106 30 91)(31 297 46 282)(32 296 47 281)(33 295 48 280)(34 294 49 279)(35 293 50 278)(36 292 51 277)(37 291 52 276)(38 290 53 275)(39 289 54 274)(40 288 55 273)(41 287 56 272)(42 286 57 271)(43 285 58 300)(44 284 59 299)(45 283 60 298)(61 344 76 359)(62 343 77 358)(63 342 78 357)(64 341 79 356)(65 340 80 355)(66 339 81 354)(67 338 82 353)(68 337 83 352)(69 336 84 351)(70 335 85 350)(71 334 86 349)(72 333 87 348)(73 332 88 347)(74 331 89 346)(75 360 90 345)(121 209 136 194)(122 208 137 193)(123 207 138 192)(124 206 139 191)(125 205 140 190)(126 204 141 189)(127 203 142 188)(128 202 143 187)(129 201 144 186)(130 200 145 185)(131 199 146 184)(132 198 147 183)(133 197 148 182)(134 196 149 181)(135 195 150 210)(151 389 166 374)(152 388 167 373)(153 387 168 372)(154 386 169 371)(155 385 170 370)(156 384 171 369)(157 383 172 368)(158 382 173 367)(159 381 174 366)(160 380 175 365)(161 379 176 364)(162 378 177 363)(163 377 178 362)(164 376 179 361)(165 375 180 390)(211 243 226 258)(212 242 227 257)(213 241 228 256)(214 270 229 255)(215 269 230 254)(216 268 231 253)(217 267 232 252)(218 266 233 251)(219 265 234 250)(220 264 235 249)(221 263 236 248)(222 262 237 247)(223 261 238 246)(224 260 239 245)(225 259 240 244)(301 426 316 441)(302 425 317 440)(303 424 318 439)(304 423 319 438)(305 422 320 437)(306 421 321 436)(307 450 322 435)(308 449 323 434)(309 448 324 433)(310 447 325 432)(311 446 326 431)(312 445 327 430)(313 444 328 429)(314 443 329 428)(315 442 330 427)(391 465 406 480)(392 464 407 479)(393 463 408 478)(394 462 409 477)(395 461 410 476)(396 460 411 475)(397 459 412 474)(398 458 413 473)(399 457 414 472)(400 456 415 471)(401 455 416 470)(402 454 417 469)(403 453 418 468)(404 452 419 467)(405 451 420 466)
(1 340 221 329)(2 341 222 330)(3 342 223 301)(4 343 224 302)(5 344 225 303)(6 345 226 304)(7 346 227 305)(8 347 228 306)(9 348 229 307)(10 349 230 308)(11 350 231 309)(12 351 232 310)(13 352 233 311)(14 353 234 312)(15 354 235 313)(16 355 236 314)(17 356 237 315)(18 357 238 316)(19 358 239 317)(20 359 240 318)(21 360 211 319)(22 331 212 320)(23 332 213 321)(24 333 214 322)(25 334 215 323)(26 335 216 324)(27 336 217 325)(28 337 218 326)(29 338 219 327)(30 339 220 328)(31 419 209 384)(32 420 210 385)(33 391 181 386)(34 392 182 387)(35 393 183 388)(36 394 184 389)(37 395 185 390)(38 396 186 361)(39 397 187 362)(40 398 188 363)(41 399 189 364)(42 400 190 365)(43 401 191 366)(44 402 192 367)(45 403 193 368)(46 404 194 369)(47 405 195 370)(48 406 196 371)(49 407 197 372)(50 408 198 373)(51 409 199 374)(52 410 200 375)(53 411 201 376)(54 412 202 377)(55 413 203 378)(56 414 204 379)(57 415 205 380)(58 416 206 381)(59 417 207 382)(60 418 208 383)(61 259 439 116)(62 260 440 117)(63 261 441 118)(64 262 442 119)(65 263 443 120)(66 264 444 91)(67 265 445 92)(68 266 446 93)(69 267 447 94)(70 268 448 95)(71 269 449 96)(72 270 450 97)(73 241 421 98)(74 242 422 99)(75 243 423 100)(76 244 424 101)(77 245 425 102)(78 246 426 103)(79 247 427 104)(80 248 428 105)(81 249 429 106)(82 250 430 107)(83 251 431 108)(84 252 432 109)(85 253 433 110)(86 254 434 111)(87 255 435 112)(88 256 436 113)(89 257 437 114)(90 258 438 115)(121 171 282 467)(122 172 283 468)(123 173 284 469)(124 174 285 470)(125 175 286 471)(126 176 287 472)(127 177 288 473)(128 178 289 474)(129 179 290 475)(130 180 291 476)(131 151 292 477)(132 152 293 478)(133 153 294 479)(134 154 295 480)(135 155 296 451)(136 156 297 452)(137 157 298 453)(138 158 299 454)(139 159 300 455)(140 160 271 456)(141 161 272 457)(142 162 273 458)(143 163 274 459)(144 164 275 460)(145 165 276 461)(146 166 277 462)(147 167 278 463)(148 168 279 464)(149 169 280 465)(150 170 281 466)
(1 409 221 374)(2 398 222 363)(3 417 223 382)(4 406 224 371)(5 395 225 390)(6 414 226 379)(7 403 227 368)(8 392 228 387)(9 411 229 376)(10 400 230 365)(11 419 231 384)(12 408 232 373)(13 397 233 362)(14 416 234 381)(15 405 235 370)(16 394 236 389)(17 413 237 378)(18 402 238 367)(19 391 239 386)(20 410 240 375)(21 399 211 364)(22 418 212 383)(23 407 213 372)(24 396 214 361)(25 415 215 380)(26 404 216 369)(27 393 217 388)(28 412 218 377)(29 401 219 366)(30 420 220 385)(31 324 209 335)(32 313 210 354)(33 302 181 343)(34 321 182 332)(35 310 183 351)(36 329 184 340)(37 318 185 359)(38 307 186 348)(39 326 187 337)(40 315 188 356)(41 304 189 345)(42 323 190 334)(43 312 191 353)(44 301 192 342)(45 320 193 331)(46 309 194 350)(47 328 195 339)(48 317 196 358)(49 306 197 347)(50 325 198 336)(51 314 199 355)(52 303 200 344)(53 322 201 333)(54 311 202 352)(55 330 203 341)(56 319 204 360)(57 308 205 349)(58 327 206 338)(59 316 207 357)(60 305 208 346)(61 291 439 130)(62 280 440 149)(63 299 441 138)(64 288 442 127)(65 277 443 146)(66 296 444 135)(67 285 445 124)(68 274 446 143)(69 293 447 132)(70 282 448 121)(71 271 449 140)(72 290 450 129)(73 279 421 148)(74 298 422 137)(75 287 423 126)(76 276 424 145)(77 295 425 134)(78 284 426 123)(79 273 427 142)(80 292 428 131)(81 281 429 150)(82 300 430 139)(83 289 431 128)(84 278 432 147)(85 297 433 136)(86 286 434 125)(87 275 435 144)(88 294 436 133)(89 283 437 122)(90 272 438 141)(91 466 264 170)(92 455 265 159)(93 474 266 178)(94 463 267 167)(95 452 268 156)(96 471 269 175)(97 460 270 164)(98 479 241 153)(99 468 242 172)(100 457 243 161)(101 476 244 180)(102 465 245 169)(103 454 246 158)(104 473 247 177)(105 462 248 166)(106 451 249 155)(107 470 250 174)(108 459 251 163)(109 478 252 152)(110 467 253 171)(111 456 254 160)(112 475 255 179)(113 464 256 168)(114 453 257 157)(115 472 258 176)(116 461 259 165)(117 480 260 154)(118 469 261 173)(119 458 262 162)(120 477 263 151)

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

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

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

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×D5D15⋊Q8D125D5Dic5.D6
kernelDic15.Q8Dic3×Dic5Dic155C4C6.Dic10C3×C10.D4C5×C4⋊Dic3C30.4Q8C10.D4Dic15C4⋊Dic3C2×Dic5C2×C20C30C2×Dic3C2×C12C10C6C10C10C2×C4C6C6C22C2C2C2
# reps11211111222144248112222444

Matrix representation of Dic15.Q8 in GL6(𝔽61)

1600000
100000
0017100
0060000
000010
000001
,
10260000
36510000
00413400
0082000
000010
000001
,
38460000
15230000
0060000
0006000
000083
00001953
,
1100000
0110000
00202700
00534100
00002733
00001334

G:=sub<GL(6,GF(61))| [1,1,0,0,0,0,60,0,0,0,0,0,0,0,17,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,36,0,0,0,0,26,51,0,0,0,0,0,0,41,8,0,0,0,0,34,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[38,15,0,0,0,0,46,23,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,8,19,0,0,0,0,3,53],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,20,53,0,0,0,0,27,41,0,0,0,0,0,0,27,13,0,0,0,0,33,34] >;

Dic15.Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}.Q_8
% in TeX

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

G:=SmallGroup(480,412);
// by ID

G=gap.SmallGroup(480,412);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,254,219,100,1356,18822]);
// Polycyclic

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

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