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G = Dic15.3Q8order 480 = 25·3·5

1st non-split extension by Dic15 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic15.3Q8, C4⋊C4.5D15, C2.5(Q8×D15), C6.40(Q8×D5), (C2×C4).11D30, (C2×C20).37D6, C10.40(S3×Q8), C30.93(C2×Q8), C605C4.8C2, (C2×C12).36D10, C57(Dic3.Q8), C6.99(C4○D20), C1516(C42.C2), C30.4Q8.3C2, C30.172(C4○D4), C10.99(C4○D12), C6.96(D42D5), (C2×C60).177C22, (C2×C30).286C23, (C4×Dic15).11C2, C37(Dic5.Q8), C2.11(D42D15), C10.96(D42S3), C2.13(D6011C2), C22.47(C22×D15), (C2×Dic15).161C22, (C5×C4⋊C4).6S3, (C3×C4⋊C4).6D5, (C15×C4⋊C4).6C2, (C2×C6).282(C22×D5), (C2×C10).281(C22×S3), SmallGroup(480,854)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic15.3Q8
C1C5C15C30C2×C30C2×Dic15C4×Dic15 — Dic15.3Q8
C15C2×C30 — Dic15.3Q8
C1C22C4⋊C4

Generators and relations for Dic15.3Q8
 G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a15b, bd=db, dcd-1=a15c-1 >

Subgroups: 516 in 112 conjugacy classes, 49 normal (47 characteristic)
C1, C2 [×3], C3, C4 [×8], C22, C5, C6 [×3], C2×C4 [×3], C2×C4 [×4], C10 [×3], Dic3 [×5], C12 [×3], C2×C6, C15, C42, C4⋊C4, C4⋊C4 [×5], Dic5 [×5], C20 [×3], C2×C10, C2×Dic3 [×4], C2×C12 [×3], C30 [×3], C42.C2, C2×Dic5 [×4], C2×C20 [×3], C4×Dic3, Dic3⋊C4 [×4], C4⋊Dic3, C3×C4⋊C4, Dic15 [×2], Dic15 [×3], C60 [×3], C2×C30, C4×Dic5, C10.D4 [×4], C4⋊Dic5, C5×C4⋊C4, Dic3.Q8, C2×Dic15 [×4], C2×C60 [×3], Dic5.Q8, C4×Dic15, C30.4Q8 [×4], C605C4, C15×C4⋊C4, Dic15.3Q8
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D5, D6 [×3], C2×Q8, C4○D4 [×2], D10 [×3], C22×S3, D15, C42.C2, C22×D5, C4○D12, D42S3, S3×Q8, D30 [×3], C4○D20, D42D5, Q8×D5, Dic3.Q8, C22×D15, Dic5.Q8, D6011C2, D42D15, Q8×D15, Dic15.3Q8

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

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

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

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

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A···10F12A···12F15A15B15C15D20A···20L30A···30L60A···60X
order1222344444444445566610···1012···121515151520···2030···3060···60
size111122244303030306060222222···24···422224···42···24···4

84 irreducible representations

dim1111122222222222444444
type++++++-+++++------
imageC1C2C2C2C2S3Q8D5D6C4○D4D10D15C4○D12D30C4○D20D6011C2D42S3S3×Q8D42D5Q8×D5D42D15Q8×D15
kernelDic15.3Q8C4×Dic15C30.4Q8C605C4C15×C4⋊C4C5×C4⋊C4Dic15C3×C4⋊C4C2×C20C30C2×C12C4⋊C4C10C2×C4C6C2C10C10C6C6C2C2
# reps114111223464412816112244

Matrix representation of Dic15.3Q8 in GL6(𝔽61)

1600000
0420000
0027000
0005200
000010
000001
,
010000
100000
0005000
0050000
000010
000001
,
100000
010000
001000
0006000
0000159
0000160
,
010000
100000
000100
001000
0000942
00003052

G:=sub<GL(6,GF(61))| [16,0,0,0,0,0,0,42,0,0,0,0,0,0,27,0,0,0,0,0,0,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,50,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,59,60],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,30,0,0,0,0,42,52] >;

Dic15.3Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}._3Q_8
% in TeX

G:=Group("Dic15.3Q8");
// GroupNames label

G:=SmallGroup(480,854);
// by ID

G=gap.SmallGroup(480,854);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,254,219,100,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=1,b^2=a^15,d^2=c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,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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