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G = Dic151Q8order 480 = 25·3·5

1st semidirect product of Dic15 and Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic151Q8, Dic31Dic10, C151(C4⋊Q8), C35(C20⋊Q8), C6.2(Q8×D5), (C2×C20).2D6, C30.4(C2×Q8), (C5×Dic3)⋊1Q8, C30.99(C2×D4), C6.124(D4×D5), C10.20(S3×Q8), C2.6(D15⋊Q8), Dic3⋊C4.7D5, (C3×Dic5).6D4, C2.6(S3×Dic10), C6.2(C2×Dic10), (C2×C12).216D10, C52(Dic3⋊Q8), (C2×C30).17C23, (C2×Dic5).81D6, (C2×Dic10).4S3, C6.Dic10.2C2, C30.4Q8.6C2, Dic155C4.2C2, (C2×C60).247C22, (Dic3×Dic5).4C2, (C6×Dic10).13C2, (C2×Dic3).73D10, (C6×Dic5).4C22, Dic5.10(C3⋊D4), (C10×Dic3).4C22, (C2×Dic15).25C22, C2.9(D5×C3⋊D4), (C2×C15⋊Q8).4C2, (C2×C4).20(S3×D5), C10.26(C2×C3⋊D4), C22.114(C2×S3×D5), (C5×Dic3⋊C4).7C2, (C2×C6).29(C22×D5), (C2×C10).29(C22×S3), SmallGroup(480,403)

Series: Derived Chief Lower central Upper central

C1C2×C30 — Dic151Q8
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic151Q8
C15C2×C30 — Dic151Q8
C1C22C2×C4

Generators and relations for Dic151Q8
 G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=a-1, ac=ca, dad-1=a19, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 604 in 136 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C3, C4 [×10], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×2], Dic3 [×4], C12 [×4], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8 [×2], Dic5 [×2], Dic5 [×4], C20 [×4], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C30 [×3], C4⋊Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3, Dic3⋊C4, Dic3⋊C4 [×3], C2×Dic6, C6×Q8, C5×Dic3 [×2], C5×Dic3, C3×Dic5 [×2], C3×Dic5, Dic15 [×2], Dic15, C60, C2×C30, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4, C2×Dic10, C2×Dic10, Dic3⋊Q8, C15⋊Q8 [×2], C3×Dic10 [×2], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, C20⋊Q8, Dic3×Dic5, Dic155C4, C6.Dic10, C5×Dic3⋊C4, C30.4Q8, C2×C15⋊Q8, C6×Dic10, Dic151Q8
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, Dic10 [×2], C22×D5, S3×Q8 [×2], C2×C3⋊D4, S3×D5, C2×Dic10, D4×D5, Q8×D5, Dic3⋊Q8, C2×S3×D5, C20⋊Q8, S3×Dic10, D15⋊Q8, D5×C3⋊D4, Dic151Q8

Smallest permutation representation of Dic151Q8
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)
(1 269 16 254)(2 268 17 253)(3 267 18 252)(4 266 19 251)(5 265 20 250)(6 264 21 249)(7 263 22 248)(8 262 23 247)(9 261 24 246)(10 260 25 245)(11 259 26 244)(12 258 27 243)(13 257 28 242)(14 256 29 241)(15 255 30 270)(31 296 46 281)(32 295 47 280)(33 294 48 279)(34 293 49 278)(35 292 50 277)(36 291 51 276)(37 290 52 275)(38 289 53 274)(39 288 54 273)(40 287 55 272)(41 286 56 271)(42 285 57 300)(43 284 58 299)(44 283 59 298)(45 282 60 297)(61 370 76 385)(62 369 77 384)(63 368 78 383)(64 367 79 382)(65 366 80 381)(66 365 81 380)(67 364 82 379)(68 363 83 378)(69 362 84 377)(70 361 85 376)(71 390 86 375)(72 389 87 374)(73 388 88 373)(74 387 89 372)(75 386 90 371)(91 240 106 225)(92 239 107 224)(93 238 108 223)(94 237 109 222)(95 236 110 221)(96 235 111 220)(97 234 112 219)(98 233 113 218)(99 232 114 217)(100 231 115 216)(101 230 116 215)(102 229 117 214)(103 228 118 213)(104 227 119 212)(105 226 120 211)(121 402 136 417)(122 401 137 416)(123 400 138 415)(124 399 139 414)(125 398 140 413)(126 397 141 412)(127 396 142 411)(128 395 143 410)(129 394 144 409)(130 393 145 408)(131 392 146 407)(132 391 147 406)(133 420 148 405)(134 419 149 404)(135 418 150 403)(151 186 166 201)(152 185 167 200)(153 184 168 199)(154 183 169 198)(155 182 170 197)(156 181 171 196)(157 210 172 195)(158 209 173 194)(159 208 174 193)(160 207 175 192)(161 206 176 191)(162 205 177 190)(163 204 178 189)(164 203 179 188)(165 202 180 187)(301 448 316 433)(302 447 317 432)(303 446 318 431)(304 445 319 430)(305 444 320 429)(306 443 321 428)(307 442 322 427)(308 441 323 426)(309 440 324 425)(310 439 325 424)(311 438 326 423)(312 437 327 422)(313 436 328 421)(314 435 329 450)(315 434 330 449)(331 452 346 467)(332 451 347 466)(333 480 348 465)(334 479 349 464)(335 478 350 463)(336 477 351 462)(337 476 352 461)(338 475 353 460)(339 474 354 459)(340 473 355 458)(341 472 356 457)(342 471 357 456)(343 470 358 455)(344 469 359 454)(345 468 360 453)
(1 272 187 434)(2 273 188 435)(3 274 189 436)(4 275 190 437)(5 276 191 438)(6 277 192 439)(7 278 193 440)(8 279 194 441)(9 280 195 442)(10 281 196 443)(11 282 197 444)(12 283 198 445)(13 284 199 446)(14 285 200 447)(15 286 201 448)(16 287 202 449)(17 288 203 450)(18 289 204 421)(19 290 205 422)(20 291 206 423)(21 292 207 424)(22 293 208 425)(23 294 209 426)(24 295 210 427)(25 296 181 428)(26 297 182 429)(27 298 183 430)(28 299 184 431)(29 300 185 432)(30 271 186 433)(31 171 321 245)(32 172 322 246)(33 173 323 247)(34 174 324 248)(35 175 325 249)(36 176 326 250)(37 177 327 251)(38 178 328 252)(39 179 329 253)(40 180 330 254)(41 151 301 255)(42 152 302 256)(43 153 303 257)(44 154 304 258)(45 155 305 259)(46 156 306 260)(47 157 307 261)(48 158 308 262)(49 159 309 263)(50 160 310 264)(51 161 311 265)(52 162 312 266)(53 163 313 267)(54 164 314 268)(55 165 315 269)(56 166 316 270)(57 167 317 241)(58 168 318 242)(59 169 319 243)(60 170 320 244)(61 411 99 455)(62 412 100 456)(63 413 101 457)(64 414 102 458)(65 415 103 459)(66 416 104 460)(67 417 105 461)(68 418 106 462)(69 419 107 463)(70 420 108 464)(71 391 109 465)(72 392 110 466)(73 393 111 467)(74 394 112 468)(75 395 113 469)(76 396 114 470)(77 397 115 471)(78 398 116 472)(79 399 117 473)(80 400 118 474)(81 401 119 475)(82 402 120 476)(83 403 91 477)(84 404 92 478)(85 405 93 479)(86 406 94 480)(87 407 95 451)(88 408 96 452)(89 409 97 453)(90 410 98 454)(121 211 337 379)(122 212 338 380)(123 213 339 381)(124 214 340 382)(125 215 341 383)(126 216 342 384)(127 217 343 385)(128 218 344 386)(129 219 345 387)(130 220 346 388)(131 221 347 389)(132 222 348 390)(133 223 349 361)(134 224 350 362)(135 225 351 363)(136 226 352 364)(137 227 353 365)(138 228 354 366)(139 229 355 367)(140 230 356 368)(141 231 357 369)(142 232 358 370)(143 233 359 371)(144 234 360 372)(145 235 331 373)(146 236 332 374)(147 237 333 375)(148 238 334 376)(149 239 335 377)(150 240 336 378)
(1 143 187 359)(2 132 188 348)(3 121 189 337)(4 140 190 356)(5 129 191 345)(6 148 192 334)(7 137 193 353)(8 126 194 342)(9 145 195 331)(10 134 196 350)(11 123 197 339)(12 142 198 358)(13 131 199 347)(14 150 200 336)(15 139 201 355)(16 128 202 344)(17 147 203 333)(18 136 204 352)(19 125 205 341)(20 144 206 360)(21 133 207 349)(22 122 208 338)(23 141 209 357)(24 130 210 346)(25 149 181 335)(26 138 182 354)(27 127 183 343)(28 146 184 332)(29 135 185 351)(30 124 186 340)(31 84 321 92)(32 73 322 111)(33 62 323 100)(34 81 324 119)(35 70 325 108)(36 89 326 97)(37 78 327 116)(38 67 328 105)(39 86 329 94)(40 75 330 113)(41 64 301 102)(42 83 302 91)(43 72 303 110)(44 61 304 99)(45 80 305 118)(46 69 306 107)(47 88 307 96)(48 77 308 115)(49 66 309 104)(50 85 310 93)(51 74 311 112)(52 63 312 101)(53 82 313 120)(54 71 314 109)(55 90 315 98)(56 79 316 117)(57 68 317 106)(58 87 318 95)(59 76 319 114)(60 65 320 103)(151 458 255 414)(152 477 256 403)(153 466 257 392)(154 455 258 411)(155 474 259 400)(156 463 260 419)(157 452 261 408)(158 471 262 397)(159 460 263 416)(160 479 264 405)(161 468 265 394)(162 457 266 413)(163 476 267 402)(164 465 268 391)(165 454 269 410)(166 473 270 399)(167 462 241 418)(168 451 242 407)(169 470 243 396)(170 459 244 415)(171 478 245 404)(172 467 246 393)(173 456 247 412)(174 475 248 401)(175 464 249 420)(176 453 250 409)(177 472 251 398)(178 461 252 417)(179 480 253 406)(180 469 254 395)(211 274 379 436)(212 293 380 425)(213 282 381 444)(214 271 382 433)(215 290 383 422)(216 279 384 441)(217 298 385 430)(218 287 386 449)(219 276 387 438)(220 295 388 427)(221 284 389 446)(222 273 390 435)(223 292 361 424)(224 281 362 443)(225 300 363 432)(226 289 364 421)(227 278 365 440)(228 297 366 429)(229 286 367 448)(230 275 368 437)(231 294 369 426)(232 283 370 445)(233 272 371 434)(234 291 372 423)(235 280 373 442)(236 299 374 431)(237 288 375 450)(238 277 376 439)(239 296 377 428)(240 285 378 447)

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

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

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

60 conjugacy classes

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

60 irreducible representations

dim111111112222222222244444444
type+++++++++-+-+++++--++-+-
imageC1C2C2C2C2C2C2C2S3Q8D4Q8D5D6D6D10D10C3⋊D4Dic10S3×Q8S3×D5D4×D5Q8×D5C2×S3×D5S3×Dic10D15⋊Q8D5×C3⋊D4
kernelDic151Q8Dic3×Dic5Dic155C4C6.Dic10C5×Dic3⋊C4C30.4Q8C2×C15⋊Q8C6×Dic10C2×Dic10C5×Dic3C3×Dic5Dic15Dic3⋊C4C2×Dic5C2×C20C2×Dic3C2×C12Dic5Dic3C10C2×C4C6C6C22C2C2C2
# reps111111111222221424822222444

Matrix representation of Dic151Q8 in GL6(𝔽61)

100000
010000
0060100
0060000
0000160
00004517
,
6000000
0600000
0016000
0006000
00002729
00001934
,
010000
6000000
0060000
0006000
0000324
0000329
,
3980000
8220000
0060000
0006000
00002729
00001934

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,1,45,0,0,0,0,60,17],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,60,60,0,0,0,0,0,0,27,19,0,0,0,0,29,34],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,32,3,0,0,0,0,4,29],[39,8,0,0,0,0,8,22,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,27,19,0,0,0,0,29,34] >;

Dic151Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_1Q_8
% in TeX

G:=Group("Dic15:1Q8");
// GroupNames label

G:=SmallGroup(480,403);
// by ID

G=gap.SmallGroup(480,403);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,254,219,58,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=1,b^2=a^15,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^19,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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