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G = Dic157Q8order 480 = 25·3·5

5th semidirect product of Dic15 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic157Q8, Dic65Dic5, C1511(C4×Q8), C32(Q8×Dic5), C6.6(Q8×D5), C10.6(S3×Q8), C20.57(C4×S3), C30.17(C2×Q8), C60.133(C2×C4), (C5×Dic6)⋊12C4, (C2×C20).108D6, C2.3(D15⋊Q8), C4⋊Dic5.11S3, (C2×Dic6).8D5, C4.10(S3×Dic5), C30.13(C4○D4), (C2×C12).110D10, C56(Dic6⋊C4), (C2×C30).34C23, (C10×Dic6).8C2, (C2×Dic5).91D6, C12.13(C2×Dic5), C6.Dic10.8C2, C10.1(D42S3), C2.1(D20⋊S3), C6.8(C22×Dic5), (C2×C60).192C22, C30.117(C22×C4), C6.20(Q82D5), Dic3.4(C2×Dic5), (C2×Dic3).84D10, (C4×Dic15).13C2, (Dic3×Dic5).17C2, (C6×Dic5).19C22, (C10×Dic3).18C22, (C2×Dic15).181C22, C10.116(S3×C2×C4), C2.10(C2×S3×Dic5), C22.30(C2×S3×D5), (C2×C4).202(S3×D5), (C3×C4⋊Dic5).10C2, (C2×C6).46(C22×D5), (C2×C10).46(C22×S3), (C5×Dic3).26(C2×C4), SmallGroup(480,420)

Series: Derived Chief Lower central Upper central

C1C30 — Dic157Q8
C1C5C15C30C2×C30C6×Dic5Dic3×Dic5 — Dic157Q8
C15C30 — Dic157Q8
C1C22C2×C4

Generators and relations for Dic157Q8
 G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=a-1, ac=ca, dad-1=a11, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 492 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 [×4], Dic3 [×3], C12 [×2], C12 [×2], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5 [×5], C20 [×2], C20 [×4], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4×Q8, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C4×Dic3 [×3], Dic3⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C5×Dic3 [×4], C3×Dic5 [×2], Dic15 [×2], Dic15, C60 [×2], C2×C30, C4×Dic5 [×3], C4⋊Dic5, C4⋊Dic5 [×2], Q8×C10, Dic6⋊C4, C6×Dic5 [×2], C5×Dic6 [×4], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, Q8×Dic5, Dic3×Dic5 [×2], C6.Dic10 [×2], C3×C4⋊Dic5, C4×Dic15, C10×Dic6, Dic157Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D5, D6 [×3], C22×C4, C2×Q8, C4○D4, Dic5 [×4], D10 [×3], C4×S3 [×2], C22×S3, C4×Q8, C2×Dic5 [×6], C22×D5, S3×C2×C4, D42S3, S3×Q8, S3×D5, Q8×D5, Q82D5, C22×Dic5, Dic6⋊C4, S3×Dic5 [×2], C2×S3×D5, Q8×Dic5, D20⋊S3, D15⋊Q8, C2×S3×Dic5, Dic157Q8

Smallest permutation representation of Dic157Q8
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)
(1 361 16 376)(2 390 17 375)(3 389 18 374)(4 388 19 373)(5 387 20 372)(6 386 21 371)(7 385 22 370)(8 384 23 369)(9 383 24 368)(10 382 25 367)(11 381 26 366)(12 380 27 365)(13 379 28 364)(14 378 29 363)(15 377 30 362)(31 173 46 158)(32 172 47 157)(33 171 48 156)(34 170 49 155)(35 169 50 154)(36 168 51 153)(37 167 52 152)(38 166 53 151)(39 165 54 180)(40 164 55 179)(41 163 56 178)(42 162 57 177)(43 161 58 176)(44 160 59 175)(45 159 60 174)(61 129 76 144)(62 128 77 143)(63 127 78 142)(64 126 79 141)(65 125 80 140)(66 124 81 139)(67 123 82 138)(68 122 83 137)(69 121 84 136)(70 150 85 135)(71 149 86 134)(72 148 87 133)(73 147 88 132)(74 146 89 131)(75 145 90 130)(91 436 106 421)(92 435 107 450)(93 434 108 449)(94 433 109 448)(95 432 110 447)(96 431 111 446)(97 430 112 445)(98 429 113 444)(99 428 114 443)(100 427 115 442)(101 426 116 441)(102 425 117 440)(103 424 118 439)(104 423 119 438)(105 422 120 437)(181 453 196 468)(182 452 197 467)(183 451 198 466)(184 480 199 465)(185 479 200 464)(186 478 201 463)(187 477 202 462)(188 476 203 461)(189 475 204 460)(190 474 205 459)(191 473 206 458)(192 472 207 457)(193 471 208 456)(194 470 209 455)(195 469 210 454)(211 336 226 351)(212 335 227 350)(213 334 228 349)(214 333 229 348)(215 332 230 347)(216 331 231 346)(217 360 232 345)(218 359 233 344)(219 358 234 343)(220 357 235 342)(221 356 236 341)(222 355 237 340)(223 354 238 339)(224 353 239 338)(225 352 240 337)(241 419 256 404)(242 418 257 403)(243 417 258 402)(244 416 259 401)(245 415 260 400)(246 414 261 399)(247 413 262 398)(248 412 263 397)(249 411 264 396)(250 410 265 395)(251 409 266 394)(252 408 267 393)(253 407 268 392)(254 406 269 391)(255 405 270 420)(271 315 286 330)(272 314 287 329)(273 313 288 328)(274 312 289 327)(275 311 290 326)(276 310 291 325)(277 309 292 324)(278 308 293 323)(279 307 294 322)(280 306 295 321)(281 305 296 320)(282 304 297 319)(283 303 298 318)(284 302 299 317)(285 301 300 316)
(1 142 258 114)(2 143 259 115)(3 144 260 116)(4 145 261 117)(5 146 262 118)(6 147 263 119)(7 148 264 120)(8 149 265 91)(9 150 266 92)(10 121 267 93)(11 122 268 94)(12 123 269 95)(13 124 270 96)(14 125 241 97)(15 126 242 98)(16 127 243 99)(17 128 244 100)(18 129 245 101)(19 130 246 102)(20 131 247 103)(21 132 248 104)(22 133 249 105)(23 134 250 106)(24 135 251 107)(25 136 252 108)(26 137 253 109)(27 138 254 110)(28 139 255 111)(29 140 256 112)(30 141 257 113)(31 463 336 310)(32 464 337 311)(33 465 338 312)(34 466 339 313)(35 467 340 314)(36 468 341 315)(37 469 342 316)(38 470 343 317)(39 471 344 318)(40 472 345 319)(41 473 346 320)(42 474 347 321)(43 475 348 322)(44 476 349 323)(45 477 350 324)(46 478 351 325)(47 479 352 326)(48 480 353 327)(49 451 354 328)(50 452 355 329)(51 453 356 330)(52 454 357 301)(53 455 358 302)(54 456 359 303)(55 457 360 304)(56 458 331 305)(57 459 332 306)(58 460 333 307)(59 461 334 308)(60 462 335 309)(61 400 441 389)(62 401 442 390)(63 402 443 361)(64 403 444 362)(65 404 445 363)(66 405 446 364)(67 406 447 365)(68 407 448 366)(69 408 449 367)(70 409 450 368)(71 410 421 369)(72 411 422 370)(73 412 423 371)(74 413 424 372)(75 414 425 373)(76 415 426 374)(77 416 427 375)(78 417 428 376)(79 418 429 377)(80 419 430 378)(81 420 431 379)(82 391 432 380)(83 392 433 381)(84 393 434 382)(85 394 435 383)(86 395 436 384)(87 396 437 385)(88 397 438 386)(89 398 439 387)(90 399 440 388)(151 194 234 299)(152 195 235 300)(153 196 236 271)(154 197 237 272)(155 198 238 273)(156 199 239 274)(157 200 240 275)(158 201 211 276)(159 202 212 277)(160 203 213 278)(161 204 214 279)(162 205 215 280)(163 206 216 281)(164 207 217 282)(165 208 218 283)(166 209 219 284)(167 210 220 285)(168 181 221 286)(169 182 222 287)(170 183 223 288)(171 184 224 289)(172 185 225 290)(173 186 226 291)(174 187 227 292)(175 188 228 293)(176 189 229 294)(177 190 230 295)(178 191 231 296)(179 192 232 297)(180 193 233 298)
(1 342 258 37)(2 353 259 48)(3 334 260 59)(4 345 261 40)(5 356 262 51)(6 337 263 32)(7 348 264 43)(8 359 265 54)(9 340 266 35)(10 351 267 46)(11 332 268 57)(12 343 269 38)(13 354 270 49)(14 335 241 60)(15 346 242 41)(16 357 243 52)(17 338 244 33)(18 349 245 44)(19 360 246 55)(20 341 247 36)(21 352 248 47)(22 333 249 58)(23 344 250 39)(24 355 251 50)(25 336 252 31)(26 347 253 42)(27 358 254 53)(28 339 255 34)(29 350 256 45)(30 331 257 56)(61 188 441 293)(62 199 442 274)(63 210 443 285)(64 191 444 296)(65 202 445 277)(66 183 446 288)(67 194 447 299)(68 205 448 280)(69 186 449 291)(70 197 450 272)(71 208 421 283)(72 189 422 294)(73 200 423 275)(74 181 424 286)(75 192 425 297)(76 203 426 278)(77 184 427 289)(78 195 428 300)(79 206 429 281)(80 187 430 292)(81 198 431 273)(82 209 432 284)(83 190 433 295)(84 201 434 276)(85 182 435 287)(86 193 436 298)(87 204 437 279)(88 185 438 290)(89 196 439 271)(90 207 440 282)(91 303 149 456)(92 314 150 467)(93 325 121 478)(94 306 122 459)(95 317 123 470)(96 328 124 451)(97 309 125 462)(98 320 126 473)(99 301 127 454)(100 312 128 465)(101 323 129 476)(102 304 130 457)(103 315 131 468)(104 326 132 479)(105 307 133 460)(106 318 134 471)(107 329 135 452)(108 310 136 463)(109 321 137 474)(110 302 138 455)(111 313 139 466)(112 324 140 477)(113 305 141 458)(114 316 142 469)(115 327 143 480)(116 308 144 461)(117 319 145 472)(118 330 146 453)(119 311 147 464)(120 322 148 475)(151 365 234 406)(152 376 235 417)(153 387 236 398)(154 368 237 409)(155 379 238 420)(156 390 239 401)(157 371 240 412)(158 382 211 393)(159 363 212 404)(160 374 213 415)(161 385 214 396)(162 366 215 407)(163 377 216 418)(164 388 217 399)(165 369 218 410)(166 380 219 391)(167 361 220 402)(168 372 221 413)(169 383 222 394)(170 364 223 405)(171 375 224 416)(172 386 225 397)(173 367 226 408)(174 378 227 419)(175 389 228 400)(176 370 229 411)(177 381 230 392)(178 362 231 403)(179 373 232 414)(180 384 233 395)

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

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

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

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+++++++-+++-++--+-+-+
imageC1C2C2C2C2C2C4S3Q8D5D6D6C4○D4Dic5D10D10C4×S3D42S3S3×Q8S3×D5Q8×D5Q82D5S3×Dic5C2×S3×D5D20⋊S3D15⋊Q8
kernelDic157Q8Dic3×Dic5C6.Dic10C3×C4⋊Dic5C4×Dic15C10×Dic6C5×Dic6C4⋊Dic5Dic15C2×Dic6C2×Dic5C2×C20C30Dic6C2×Dic3C2×C12C20C10C10C2×C4C6C6C4C22C2C2
# reps12211181222128424112224244

Matrix representation of Dic157Q8 in GL6(𝔽61)

59460000
4910000
00186000
001000
000010
000001
,
52590000
4090000
0072300
00275400
000010
000001
,
6000000
0600000
0060000
0006000
00005042
0000011
,
920000
21520000
0060000
0006000
00001231
00001549

G:=sub<GL(6,GF(61))| [59,49,0,0,0,0,46,1,0,0,0,0,0,0,18,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[52,40,0,0,0,0,59,9,0,0,0,0,0,0,7,27,0,0,0,0,23,54,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,42,11],[9,21,0,0,0,0,2,52,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,12,15,0,0,0,0,31,49] >;

Dic157Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_7Q_8
% in TeX

G:=Group("Dic15:7Q8");
// GroupNames label

G:=SmallGroup(480,420);
// by ID

G=gap.SmallGroup(480,420);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,219,100,1356,18822]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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