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G = Dic15.4Q8order 480 = 25·3·5

2nd non-split extension by Dic15 of Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic15.4Q8, C6.8(Q8×D5), C10.8(S3×Q8), C30.29(C2×Q8), (C2×C20).187D6, Dic3⋊C4.6D5, C55(Dic3.Q8), C6.31(C4○D20), (C2×C12).186D10, C2.11(D15⋊Q8), (C2×C30).72C23, (C2×Dic5).23D6, (C4×Dic15).8C2, C1510(C42.C2), C10.D4.6S3, C10.35(C4○D12), C30.108(C4○D4), C6.67(D42D5), (C2×C60).162C22, (C2×Dic3).22D10, C35(Dic5.Q8), C30.Q8.14C2, Dic155C4.13C2, C6.Dic10.12C2, C10.68(D42S3), (C6×Dic5).42C22, C2.14(C30.C23), C2.24(D6.D10), (C10×Dic3).41C22, (C2×Dic15).192C22, (C2×C4).175(S3×D5), C22.158(C2×S3×D5), (C5×Dic3⋊C4).6C2, (C2×C6).84(C22×D5), (C2×C10).84(C22×S3), (C3×C10.D4).6C2, SmallGroup(480,458)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic15.4Q8
C1C5C15C30C2×C30C6×Dic5Dic155C4 — Dic15.4Q8
C15C2×C30 — Dic15.4Q8
C1C22C2×C4

Generators and relations for Dic15.4Q8
 G = < a,b,c,d | a30=c4=1, b2=a15, d2=a15c2, bab-1=a-1, ac=ca, dad-1=a19, bc=cb, dbd-1=a15b, 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, Dic3⋊C4 [×3], 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, C30.Q8, Dic155C4 [×2], C6.Dic10, C3×C10.D4, C5×Dic3⋊C4, C4×Dic15, Dic15.4Q8
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, D6.D10, C30.C23, Dic15.4Q8

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

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

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

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

60 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444445566610···1012121212121215152020202020···2030···3060···60
size11112221212202030303030222222···2442020202044444412···124···44···4

60 irreducible representations

dim11111112222222222444444444
type++++++++-+++++--+--+-
imageC1C2C2C2C2C2C2S3Q8D5D6D6C4○D4D10D10C4○D12C4○D20D42S3S3×Q8S3×D5D42D5Q8×D5C2×S3×D5D15⋊Q8D6.D10C30.C23
kernelDic15.4Q8C30.Q8Dic155C4C6.Dic10C3×C10.D4C5×Dic3⋊C4C4×Dic15C10.D4Dic15Dic3⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C10C6C10C10C2×C4C6C6C22C2C2C2
# reps11211111222144248112222444

Matrix representation of Dic15.4Q8 in GL6(𝔽61)

0600000
1440000
001500
00365900
000010
000001
,
53510000
3780000
00433300
0051800
0000600
0000060
,
5000000
0500000
001000
000100
00006010
0000121
,
28370000
25330000
0060000
0006000
00002154
00003740

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,44,0,0,0,0,0,0,1,36,0,0,0,0,5,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[53,37,0,0,0,0,51,8,0,0,0,0,0,0,43,5,0,0,0,0,33,18,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,12,0,0,0,0,10,1],[28,25,0,0,0,0,37,33,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,21,37,0,0,0,0,54,40] >;

Dic15.4Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}._4Q_8
% in TeX

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

G:=SmallGroup(480,458);
// by ID

G=gap.SmallGroup(480,458);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,422,219,142,1356,18822]);
// 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,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^15*c^-1>;
// generators/relations

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