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G = Dic156Q8order 480 = 25·3·5

4th semidirect product of Dic15 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic156Q8, Dic108Dic3, C157(C4×Q8), C53(Q8×Dic3), C6.3(Q8×D5), C30.8(C2×Q8), C10.3(S3×Q8), C12.24(C4×D5), C60.132(C2×C4), C30.3(C4○D4), (C2×C20).105D6, C2.2(D15⋊Q8), C4⋊Dic3.11D5, C4.10(D5×Dic3), (C3×Dic10)⋊12C4, (C2×C12).107D10, C6.2(D42D5), C35(Dic53Q8), (C2×C30).21C23, (C2×Dic5).85D6, (C6×Dic10).8C2, (C2×Dic10).9S3, C20.33(C2×Dic3), C30.Q8.5C2, C2.1(D12⋊D5), C30.112(C22×C4), (C2×C60).191C22, (Dic3×Dic5).8C2, (C4×Dic15).12C2, (C2×Dic3).76D10, Dic5.6(C2×Dic3), (C6×Dic5).8C22, C10.20(Q83S3), (C10×Dic3).7C22, C10.20(C22×Dic3), (C2×Dic15).180C22, C6.83(C2×C4×D5), C2.9(C2×D5×Dic3), C22.25(C2×S3×D5), (C2×C4).201(S3×D5), (C5×C4⋊Dic3).10C2, (C2×C6).33(C22×D5), (C2×C10).33(C22×S3), (C3×Dic5).10(C2×C4), SmallGroup(480,407)

Series: Derived Chief Lower central Upper central

C1C30 — Dic156Q8
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic156Q8
C15C30 — Dic156Q8
C1C22C2×C4

Generators and relations for Dic156Q8
 G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=a-1, ac=ca, dad-1=a19, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 508 in 140 conjugacy classes, 70 normal (32 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×9], C22, C5, C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], Dic3 [×5], C12 [×2], C12 [×4], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×4], Dic5 [×3], C20 [×2], C20 [×2], C2×C10, C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C30 [×3], C4×Q8, Dic10 [×4], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Dic3 [×3], C4⋊Dic3, C4⋊Dic3 [×2], C6×Q8, C5×Dic3 [×2], C3×Dic5 [×4], Dic15 [×2], Dic15, C60 [×2], C2×C30, C4×Dic5 [×3], C10.D4 [×2], C5×C4⋊C4, C2×Dic10, Q8×Dic3, C3×Dic10 [×4], C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, Dic53Q8, Dic3×Dic5 [×2], C30.Q8 [×2], C5×C4⋊Dic3, C4×Dic15, C6×Dic10, Dic156Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D5, Dic3 [×4], D6 [×3], C22×C4, C2×Q8, C4○D4, D10 [×3], C2×Dic3 [×6], C22×S3, C4×Q8, C4×D5 [×2], C22×D5, S3×Q8, Q83S3, C22×Dic3, S3×D5, C2×C4×D5, D42D5, Q8×D5, Q8×Dic3, D5×Dic3 [×2], C2×S3×D5, Dic53Q8, D12⋊D5, D15⋊Q8, C2×D5×Dic3, Dic156Q8

Smallest permutation representation of Dic156Q8
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)(31 218 46 233)(32 217 47 232)(33 216 48 231)(34 215 49 230)(35 214 50 229)(36 213 51 228)(37 212 52 227)(38 211 53 226)(39 240 54 225)(40 239 55 224)(41 238 56 223)(42 237 57 222)(43 236 58 221)(44 235 59 220)(45 234 60 219)(61 169 76 154)(62 168 77 153)(63 167 78 152)(64 166 79 151)(65 165 80 180)(66 164 81 179)(67 163 82 178)(68 162 83 177)(69 161 84 176)(70 160 85 175)(71 159 86 174)(72 158 87 173)(73 157 88 172)(74 156 89 171)(75 155 90 170)(91 309 106 324)(92 308 107 323)(93 307 108 322)(94 306 109 321)(95 305 110 320)(96 304 111 319)(97 303 112 318)(98 302 113 317)(99 301 114 316)(100 330 115 315)(101 329 116 314)(102 328 117 313)(103 327 118 312)(104 326 119 311)(105 325 120 310)(121 286 136 271)(122 285 137 300)(123 284 138 299)(124 283 139 298)(125 282 140 297)(126 281 141 296)(127 280 142 295)(128 279 143 294)(129 278 144 293)(130 277 145 292)(131 276 146 291)(132 275 147 290)(133 274 148 289)(134 273 149 288)(135 272 150 287)(181 476 196 461)(182 475 197 460)(183 474 198 459)(184 473 199 458)(185 472 200 457)(186 471 201 456)(187 470 202 455)(188 469 203 454)(189 468 204 453)(190 467 205 452)(191 466 206 451)(192 465 207 480)(193 464 208 479)(194 463 209 478)(195 462 210 477)(331 388 346 373)(332 387 347 372)(333 386 348 371)(334 385 349 370)(335 384 350 369)(336 383 351 368)(337 382 352 367)(338 381 353 366)(339 380 354 365)(340 379 355 364)(341 378 356 363)(342 377 357 362)(343 376 358 361)(344 375 359 390)(345 374 360 389)(391 447 406 432)(392 446 407 431)(393 445 408 430)(394 444 409 429)(395 443 410 428)(396 442 411 427)(397 441 412 426)(398 440 413 425)(399 439 414 424)(400 438 415 423)(401 437 416 422)(402 436 417 421)(403 435 418 450)(404 434 419 449)(405 433 420 448)
(1 195 69 142)(2 196 70 143)(3 197 71 144)(4 198 72 145)(5 199 73 146)(6 200 74 147)(7 201 75 148)(8 202 76 149)(9 203 77 150)(10 204 78 121)(11 205 79 122)(12 206 80 123)(13 207 81 124)(14 208 82 125)(15 209 83 126)(16 210 84 127)(17 181 85 128)(18 182 86 129)(19 183 87 130)(20 184 88 131)(21 185 89 132)(22 186 90 133)(23 187 61 134)(24 188 62 135)(25 189 63 136)(26 190 64 137)(27 191 65 138)(28 192 66 139)(29 193 67 140)(30 194 68 141)(31 363 117 433)(32 364 118 434)(33 365 119 435)(34 366 120 436)(35 367 91 437)(36 368 92 438)(37 369 93 439)(38 370 94 440)(39 371 95 441)(40 372 96 442)(41 373 97 443)(42 374 98 444)(43 375 99 445)(44 376 100 446)(45 377 101 447)(46 378 102 448)(47 379 103 449)(48 380 104 450)(49 381 105 421)(50 382 106 422)(51 383 107 423)(52 384 108 424)(53 385 109 425)(54 386 110 426)(55 387 111 427)(56 388 112 428)(57 389 113 429)(58 390 114 430)(59 361 115 431)(60 362 116 432)(151 285 267 452)(152 286 268 453)(153 287 269 454)(154 288 270 455)(155 289 241 456)(156 290 242 457)(157 291 243 458)(158 292 244 459)(159 293 245 460)(160 294 246 461)(161 295 247 462)(162 296 248 463)(163 297 249 464)(164 298 250 465)(165 299 251 466)(166 300 252 467)(167 271 253 468)(168 272 254 469)(169 273 255 470)(170 274 256 471)(171 275 257 472)(172 276 258 473)(173 277 259 474)(174 278 260 475)(175 279 261 476)(176 280 262 477)(177 281 263 478)(178 282 264 479)(179 283 265 480)(180 284 266 451)(211 334 306 413)(212 335 307 414)(213 336 308 415)(214 337 309 416)(215 338 310 417)(216 339 311 418)(217 340 312 419)(218 341 313 420)(219 342 314 391)(220 343 315 392)(221 344 316 393)(222 345 317 394)(223 346 318 395)(224 347 319 396)(225 348 320 397)(226 349 321 398)(227 350 322 399)(228 351 323 400)(229 352 324 401)(230 353 325 402)(231 354 326 403)(232 355 327 404)(233 356 328 405)(234 357 329 406)(235 358 330 407)(236 359 301 408)(237 360 302 409)(238 331 303 410)(239 332 304 411)(240 333 305 412)
(1 325 69 230)(2 314 70 219)(3 303 71 238)(4 322 72 227)(5 311 73 216)(6 330 74 235)(7 319 75 224)(8 308 76 213)(9 327 77 232)(10 316 78 221)(11 305 79 240)(12 324 80 229)(13 313 81 218)(14 302 82 237)(15 321 83 226)(16 310 84 215)(17 329 85 234)(18 318 86 223)(19 307 87 212)(20 326 88 231)(21 315 89 220)(22 304 90 239)(23 323 61 228)(24 312 62 217)(25 301 63 236)(26 320 64 225)(27 309 65 214)(28 328 66 233)(29 317 67 222)(30 306 68 211)(31 250 117 164)(32 269 118 153)(33 258 119 172)(34 247 120 161)(35 266 91 180)(36 255 92 169)(37 244 93 158)(38 263 94 177)(39 252 95 166)(40 241 96 155)(41 260 97 174)(42 249 98 163)(43 268 99 152)(44 257 100 171)(45 246 101 160)(46 265 102 179)(47 254 103 168)(48 243 104 157)(49 262 105 176)(50 251 106 165)(51 270 107 154)(52 259 108 173)(53 248 109 162)(54 267 110 151)(55 256 111 170)(56 245 112 159)(57 264 113 178)(58 253 114 167)(59 242 115 156)(60 261 116 175)(121 393 204 344)(122 412 205 333)(123 401 206 352)(124 420 207 341)(125 409 208 360)(126 398 209 349)(127 417 210 338)(128 406 181 357)(129 395 182 346)(130 414 183 335)(131 403 184 354)(132 392 185 343)(133 411 186 332)(134 400 187 351)(135 419 188 340)(136 408 189 359)(137 397 190 348)(138 416 191 337)(139 405 192 356)(140 394 193 345)(141 413 194 334)(142 402 195 353)(143 391 196 342)(144 410 197 331)(145 399 198 350)(146 418 199 339)(147 407 200 358)(148 396 201 347)(149 415 202 336)(150 404 203 355)(271 430 468 390)(272 449 469 379)(273 438 470 368)(274 427 471 387)(275 446 472 376)(276 435 473 365)(277 424 474 384)(278 443 475 373)(279 432 476 362)(280 421 477 381)(281 440 478 370)(282 429 479 389)(283 448 480 378)(284 437 451 367)(285 426 452 386)(286 445 453 375)(287 434 454 364)(288 423 455 383)(289 442 456 372)(290 431 457 361)(291 450 458 380)(292 439 459 369)(293 428 460 388)(294 447 461 377)(295 436 462 366)(296 425 463 385)(297 444 464 374)(298 433 465 363)(299 422 466 382)(300 441 467 371)

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

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

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

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B6A6B6C10A···10F12A12B12C12D12E12F15A15B20A20B20C20D20E···20L30A···30F60A···60H
order1222344444444444444445566610···1012121212121215152020202020···2030···3060···60
size1111222666610101010151515153030222222···2442020202044444412···124···44···4

66 irreducible representations

dim11111112222222222444444444
type+++++++-+-++++-++---+
imageC1C2C2C2C2C2C4S3Q8D5Dic3D6D6C4○D4D10D10C4×D5S3×Q8Q83S3S3×D5D42D5Q8×D5D5×Dic3C2×S3×D5D12⋊D5D15⋊Q8
kernelDic156Q8Dic3×Dic5C30.Q8C5×C4⋊Dic3C4×Dic15C6×Dic10C3×Dic10C2×Dic10Dic15C4⋊Dic3Dic10C2×Dic5C2×C20C30C2×Dic3C2×C12C12C10C10C2×C4C6C6C4C22C2C2
# reps12211181224212428112224244

Matrix representation of Dic156Q8 in GL7(𝔽61)

60000000
0100000
0010000
00044100
000166000
00000601
00000600
,
50000000
0100000
0010000
000552200
00040600
000003451
000002427
,
1000000
0010000
06000000
0001000
0000100
0000010
0000001
,
1000000
022530000
053390000
000552200
00040600
0000010
0000001

G:=sub<GL(7,GF(61))| [60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,44,16,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,60,0,0,0,0,0,1,0],[50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,55,40,0,0,0,0,0,22,6,0,0,0,0,0,0,0,34,24,0,0,0,0,0,51,27],[1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,22,53,0,0,0,0,0,53,39,0,0,0,0,0,0,0,55,40,0,0,0,0,0,22,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1] >;

Dic156Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_6Q_8
% in TeX

G:=Group("Dic15:6Q8");
// GroupNames label

G:=SmallGroup(480,407);
// by ID

G=gap.SmallGroup(480,407);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,422,219,100,1356,18822]);
// 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^19,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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