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G = Dic158Q8order 480 = 25·3·5

6th semidirect product of Dic15 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.91D4, Dic158Q8, C155(C4⋊Q8), C6.10(Q8×D5), C30.31(C2×Q8), C10.10(S3×Q8), C30.121(C2×D4), (C2×C20).120D6, (C2×Dic6).9D5, (C2×C12).121D10, C53(Dic3⋊Q8), C33(Dic5⋊Q8), C12.40(C5⋊D4), C20.37(C3⋊D4), C4.11(C15⋊D4), C2.13(D15⋊Q8), (C2×C30).75C23, (C6×Dic10).9C2, (C10×Dic6).9C2, (C2×Dic5).26D6, (C2×C60).198C22, (C2×Dic3).25D10, (C2×Dic10).10S3, (C4×Dic15).15C2, Dic155C4.14C2, (C6×Dic5).45C22, (C10×Dic3).44C22, (C2×Dic15).195C22, C6.84(C2×C5⋊D4), (C2×C4).208(S3×D5), C10.85(C2×C3⋊D4), C2.18(C2×C15⋊D4), C22.161(C2×S3×D5), (C2×C6).87(C22×D5), (C2×C10).87(C22×S3), SmallGroup(480,461)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic158Q8
C1C5C15C30C2×C30C6×Dic5Dic155C4 — Dic158Q8
C15C2×C30 — Dic158Q8
C1C22C2×C4

Generators and relations for Dic158Q8
 G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=c-1 >

Subgroups: 540 in 136 conjugacy classes, 56 normal (22 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×8], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×6], Q8 [×4], C10, C10 [×2], Dic3 [×6], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×6], C20 [×2], C20 [×2], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C30, C30 [×2], C4⋊Q8, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C4×Dic3, Dic3⋊C4 [×4], C2×Dic6, C6×Q8, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×4], C60 [×2], C2×C30, C4×Dic5, C10.D4 [×4], C2×Dic10, Q8×C10, Dic3⋊Q8, C3×Dic10 [×2], C6×Dic5 [×2], C5×Dic6 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, Dic5⋊Q8, Dic155C4 [×4], C4×Dic15, C6×Dic10, C10×Dic6, Dic158Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D5, D6 [×3], C2×D4, C2×Q8 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, C4⋊Q8, C5⋊D4 [×2], C22×D5, S3×Q8 [×2], C2×C3⋊D4, S3×D5, Q8×D5 [×2], C2×C5⋊D4, Dic3⋊Q8, C15⋊D4 [×2], C2×S3×D5, Dic5⋊Q8, D15⋊Q8 [×2], C2×C15⋊D4, Dic158Q8

Smallest permutation representation of Dic158Q8
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)
(1 171 16 156)(2 170 17 155)(3 169 18 154)(4 168 19 153)(5 167 20 152)(6 166 21 151)(7 165 22 180)(8 164 23 179)(9 163 24 178)(10 162 25 177)(11 161 26 176)(12 160 27 175)(13 159 28 174)(14 158 29 173)(15 157 30 172)(31 294 46 279)(32 293 47 278)(33 292 48 277)(34 291 49 276)(35 290 50 275)(36 289 51 274)(37 288 52 273)(38 287 53 272)(39 286 54 271)(40 285 55 300)(41 284 56 299)(42 283 57 298)(43 282 58 297)(44 281 59 296)(45 280 60 295)(61 471 76 456)(62 470 77 455)(63 469 78 454)(64 468 79 453)(65 467 80 452)(66 466 81 451)(67 465 82 480)(68 464 83 479)(69 463 84 478)(70 462 85 477)(71 461 86 476)(72 460 87 475)(73 459 88 474)(74 458 89 473)(75 457 90 472)(91 311 106 326)(92 310 107 325)(93 309 108 324)(94 308 109 323)(95 307 110 322)(96 306 111 321)(97 305 112 320)(98 304 113 319)(99 303 114 318)(100 302 115 317)(101 301 116 316)(102 330 117 315)(103 329 118 314)(104 328 119 313)(105 327 120 312)(121 384 136 369)(122 383 137 368)(123 382 138 367)(124 381 139 366)(125 380 140 365)(126 379 141 364)(127 378 142 363)(128 377 143 362)(129 376 144 361)(130 375 145 390)(131 374 146 389)(132 373 147 388)(133 372 148 387)(134 371 149 386)(135 370 150 385)(181 435 196 450)(182 434 197 449)(183 433 198 448)(184 432 199 447)(185 431 200 446)(186 430 201 445)(187 429 202 444)(188 428 203 443)(189 427 204 442)(190 426 205 441)(191 425 206 440)(192 424 207 439)(193 423 208 438)(194 422 209 437)(195 421 210 436)(211 419 226 404)(212 418 227 403)(213 417 228 402)(214 416 229 401)(215 415 230 400)(216 414 231 399)(217 413 232 398)(218 412 233 397)(219 411 234 396)(220 410 235 395)(221 409 236 394)(222 408 237 393)(223 407 238 392)(224 406 239 391)(225 405 240 420)(241 346 256 331)(242 345 257 360)(243 344 258 359)(244 343 259 358)(245 342 260 357)(246 341 261 356)(247 340 262 355)(248 339 263 354)(249 338 264 353)(250 337 265 352)(251 336 266 351)(252 335 267 350)(253 334 268 349)(254 333 269 348)(255 332 270 347)
(1 335 323 238)(2 336 324 239)(3 337 325 240)(4 338 326 211)(5 339 327 212)(6 340 328 213)(7 341 329 214)(8 342 330 215)(9 343 301 216)(10 344 302 217)(11 345 303 218)(12 346 304 219)(13 347 305 220)(14 348 306 221)(15 349 307 222)(16 350 308 223)(17 351 309 224)(18 352 310 225)(19 353 311 226)(20 354 312 227)(21 355 313 228)(22 356 314 229)(23 357 315 230)(24 358 316 231)(25 359 317 232)(26 360 318 233)(27 331 319 234)(28 332 320 235)(29 333 321 236)(30 334 322 237)(31 209 83 143)(32 210 84 144)(33 181 85 145)(34 182 86 146)(35 183 87 147)(36 184 88 148)(37 185 89 149)(38 186 90 150)(39 187 61 121)(40 188 62 122)(41 189 63 123)(42 190 64 124)(43 191 65 125)(44 192 66 126)(45 193 67 127)(46 194 68 128)(47 195 69 129)(48 196 70 130)(49 197 71 131)(50 198 72 132)(51 199 73 133)(52 200 74 134)(53 201 75 135)(54 202 76 136)(55 203 77 137)(56 204 78 138)(57 205 79 139)(58 206 80 140)(59 207 81 141)(60 208 82 142)(91 419 168 264)(92 420 169 265)(93 391 170 266)(94 392 171 267)(95 393 172 268)(96 394 173 269)(97 395 174 270)(98 396 175 241)(99 397 176 242)(100 398 177 243)(101 399 178 244)(102 400 179 245)(103 401 180 246)(104 402 151 247)(105 403 152 248)(106 404 153 249)(107 405 154 250)(108 406 155 251)(109 407 156 252)(110 408 157 253)(111 409 158 254)(112 410 159 255)(113 411 160 256)(114 412 161 257)(115 413 162 258)(116 414 163 259)(117 415 164 260)(118 416 165 261)(119 417 166 262)(120 418 167 263)(271 444 456 369)(272 445 457 370)(273 446 458 371)(274 447 459 372)(275 448 460 373)(276 449 461 374)(277 450 462 375)(278 421 463 376)(279 422 464 377)(280 423 465 378)(281 424 466 379)(282 425 467 380)(283 426 468 381)(284 427 469 382)(285 428 470 383)(286 429 471 384)(287 430 472 385)(288 431 473 386)(289 432 474 387)(290 433 475 388)(291 434 476 389)(292 435 477 390)(293 436 478 361)(294 437 479 362)(295 438 480 363)(296 439 451 364)(297 440 452 365)(298 441 453 366)(299 442 454 367)(300 443 455 368)
(1 274 323 459)(2 285 324 470)(3 296 325 451)(4 277 326 462)(5 288 327 473)(6 299 328 454)(7 280 329 465)(8 291 330 476)(9 272 301 457)(10 283 302 468)(11 294 303 479)(12 275 304 460)(13 286 305 471)(14 297 306 452)(15 278 307 463)(16 289 308 474)(17 300 309 455)(18 281 310 466)(19 292 311 477)(20 273 312 458)(21 284 313 469)(22 295 314 480)(23 276 315 461)(24 287 316 472)(25 298 317 453)(26 279 318 464)(27 290 319 475)(28 271 320 456)(29 282 321 467)(30 293 322 478)(31 114 83 161)(32 95 84 172)(33 106 85 153)(34 117 86 164)(35 98 87 175)(36 109 88 156)(37 120 89 167)(38 101 90 178)(39 112 61 159)(40 93 62 170)(41 104 63 151)(42 115 64 162)(43 96 65 173)(44 107 66 154)(45 118 67 165)(46 99 68 176)(47 110 69 157)(48 91 70 168)(49 102 71 179)(50 113 72 160)(51 94 73 171)(52 105 74 152)(53 116 75 163)(54 97 76 174)(55 108 77 155)(56 119 78 166)(57 100 79 177)(58 111 80 158)(59 92 81 169)(60 103 82 180)(121 410 187 255)(122 391 188 266)(123 402 189 247)(124 413 190 258)(125 394 191 269)(126 405 192 250)(127 416 193 261)(128 397 194 242)(129 408 195 253)(130 419 196 264)(131 400 197 245)(132 411 198 256)(133 392 199 267)(134 403 200 248)(135 414 201 259)(136 395 202 270)(137 406 203 251)(138 417 204 262)(139 398 205 243)(140 409 206 254)(141 420 207 265)(142 401 208 246)(143 412 209 257)(144 393 210 268)(145 404 181 249)(146 415 182 260)(147 396 183 241)(148 407 184 252)(149 418 185 263)(150 399 186 244)(211 450 338 375)(212 431 339 386)(213 442 340 367)(214 423 341 378)(215 434 342 389)(216 445 343 370)(217 426 344 381)(218 437 345 362)(219 448 346 373)(220 429 347 384)(221 440 348 365)(222 421 349 376)(223 432 350 387)(224 443 351 368)(225 424 352 379)(226 435 353 390)(227 446 354 371)(228 427 355 382)(229 438 356 363)(230 449 357 374)(231 430 358 385)(232 441 359 366)(233 422 360 377)(234 433 331 388)(235 444 332 369)(236 425 333 380)(237 436 334 361)(238 447 335 372)(239 428 336 383)(240 439 337 364)

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

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

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

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

dim111112222222222444444
type++++++-++++++-+--+
imageC1C2C2C2C2S3Q8D4D5D6D6D10D10C3⋊D4C5⋊D4S3×Q8S3×D5Q8×D5C15⋊D4C2×S3×D5D15⋊Q8
kernelDic158Q8Dic155C4C4×Dic15C6×Dic10C10×Dic6C2×Dic10Dic15C60C2×Dic6C2×Dic5C2×C20C2×Dic3C2×C12C20C12C10C2×C4C6C4C22C2
# reps141111422214248224428

Matrix representation of Dic158Q8 in GL6(𝔽61)

2410000
52600000
000100
00604300
000010
000001
,
4470000
37170000
00594700
0022200
000010
000001
,
100000
010000
001000
000100
0000195
00001342
,
7470000
47540000
0060000
0006000
0000504
0000011

G:=sub<GL(6,GF(61))| [2,52,0,0,0,0,41,60,0,0,0,0,0,0,0,60,0,0,0,0,1,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[44,37,0,0,0,0,7,17,0,0,0,0,0,0,59,22,0,0,0,0,47,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,13,0,0,0,0,5,42],[7,47,0,0,0,0,47,54,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,4,11] >;

Dic158Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_8Q_8
% in TeX

G:=Group("Dic15:8Q8");
// GroupNames label

G:=SmallGroup(480,461);
// by ID

G=gap.SmallGroup(480,461);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,422,219,100,1356,18822]);
// 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^11,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations

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