metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊2C8, C30.5M4(2), Dic5.6Dic6, C15⋊2(C4⋊C8), C10.4(S3×C8), C30.9(C2×C8), C30.7(C4⋊C4), C6.5(D5⋊C8), C5⋊2(Dic3⋊C8), C6.14(C4⋊F5), C2.5(D15⋊C8), (C2×Dic3).5F5, (C3×Dic5).6Q8, C2.3(D6.F5), C10.5(C8⋊S3), C22.16(S3×F5), C3⋊2(Dic5⋊C8), C2.3(Dic3⋊F5), (C2×Dic15).8C4, (C10×Dic3).7C4, (C3×Dic5).35D4, C6.3(C22.F5), C10.7(Dic3⋊C4), (C2×Dic5).146D6, (Dic3×Dic5).22C2, Dic5.19(C3⋊D4), (C6×Dic5).141C22, (C6×C5⋊C8).4C2, (C2×C5⋊C8).3S3, (C2×C6).17(C2×F5), (C2×C15⋊C8).4C2, (C2×C30).11(C2×C4), (C2×C10).13(C4×S3), SmallGroup(480,253)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊C8
G = < a,b,c | a30=c8=1, b2=a15, bab-1=a-1, cac-1=a13, cbc-1=a15b >
Subgroups: 308 in 76 conjugacy classes, 36 normal (34 characteristic)
C1, C2, C3, C4, C22, C5, C6, C8, C2×C4, C10, Dic3, C12, C2×C6, C15, C42, C2×C8, Dic5, Dic5, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C30, C4⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, C5×Dic3, C3×Dic5, Dic15, C2×C30, C4×Dic5, C2×C5⋊C8, C2×C5⋊C8, Dic3⋊C8, C3×C5⋊C8, C15⋊C8, C6×Dic5, C10×Dic3, C2×Dic15, Dic5⋊C8, Dic3×Dic5, C6×C5⋊C8, C2×C15⋊C8, Dic15⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, D6, C4⋊C4, C2×C8, M4(2), F5, Dic6, C4×S3, C3⋊D4, C4⋊C8, C2×F5, S3×C8, C8⋊S3, Dic3⋊C4, D5⋊C8, C4⋊F5, C22.F5, Dic3⋊C8, S3×F5, Dic5⋊C8, Dic3⋊F5, D15⋊C8, D6.F5, Dic15⋊C8
►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)
(1 475 16 460)(2 474 17 459)(3 473 18 458)(4 472 19 457)(5 471 20 456)(6 470 21 455)(7 469 22 454)(8 468 23 453)(9 467 24 452)(10 466 25 451)(11 465 26 480)(12 464 27 479)(13 463 28 478)(14 462 29 477)(15 461 30 476)(31 407 46 392)(32 406 47 391)(33 405 48 420)(34 404 49 419)(35 403 50 418)(36 402 51 417)(37 401 52 416)(38 400 53 415)(39 399 54 414)(40 398 55 413)(41 397 56 412)(42 396 57 411)(43 395 58 410)(44 394 59 409)(45 393 60 408)(61 332 76 347)(62 331 77 346)(63 360 78 345)(64 359 79 344)(65 358 80 343)(66 357 81 342)(67 356 82 341)(68 355 83 340)(69 354 84 339)(70 353 85 338)(71 352 86 337)(72 351 87 336)(73 350 88 335)(74 349 89 334)(75 348 90 333)(91 188 106 203)(92 187 107 202)(93 186 108 201)(94 185 109 200)(95 184 110 199)(96 183 111 198)(97 182 112 197)(98 181 113 196)(99 210 114 195)(100 209 115 194)(101 208 116 193)(102 207 117 192)(103 206 118 191)(104 205 119 190)(105 204 120 189)(121 296 136 281)(122 295 137 280)(123 294 138 279)(124 293 139 278)(125 292 140 277)(126 291 141 276)(127 290 142 275)(128 289 143 274)(129 288 144 273)(130 287 145 272)(131 286 146 271)(132 285 147 300)(133 284 148 299)(134 283 149 298)(135 282 150 297)(151 266 166 251)(152 265 167 250)(153 264 168 249)(154 263 169 248)(155 262 170 247)(156 261 171 246)(157 260 172 245)(158 259 173 244)(159 258 174 243)(160 257 175 242)(161 256 176 241)(162 255 177 270)(163 254 178 269)(164 253 179 268)(165 252 180 267)(211 436 226 421)(212 435 227 450)(213 434 228 449)(214 433 229 448)(215 432 230 447)(216 431 231 446)(217 430 232 445)(218 429 233 444)(219 428 234 443)(220 427 235 442)(221 426 236 441)(222 425 237 440)(223 424 238 439)(224 423 239 438)(225 422 240 437)(301 383 316 368)(302 382 317 367)(303 381 318 366)(304 380 319 365)(305 379 320 364)(306 378 321 363)(307 377 322 362)(308 376 323 361)(309 375 324 390)(310 374 325 389)(311 373 326 388)(312 372 327 387)(313 371 328 386)(314 370 329 385)(315 369 330 384)
(1 40 180 82 184 389 135 443)(2 47 169 65 185 366 124 426)(3 54 158 78 186 373 143 439)(4 31 177 61 187 380 132 422)(5 38 166 74 188 387 121 435)(6 45 155 87 189 364 140 448)(7 52 174 70 190 371 129 431)(8 59 163 83 191 378 148 444)(9 36 152 66 192 385 137 427)(10 43 171 79 193 362 126 440)(11 50 160 62 194 369 145 423)(12 57 179 75 195 376 134 436)(13 34 168 88 196 383 123 449)(14 41 157 71 197 390 142 432)(15 48 176 84 198 367 131 445)(16 55 165 67 199 374 150 428)(17 32 154 80 200 381 139 441)(18 39 173 63 201 388 128 424)(19 46 162 76 202 365 147 437)(20 53 151 89 203 372 136 450)(21 60 170 72 204 379 125 433)(22 37 159 85 205 386 144 446)(23 44 178 68 206 363 133 429)(24 51 167 81 207 370 122 442)(25 58 156 64 208 377 141 425)(26 35 175 77 209 384 130 438)(27 42 164 90 210 361 149 421)(28 49 153 73 181 368 138 434)(29 56 172 86 182 375 127 447)(30 33 161 69 183 382 146 430)(91 312 281 227 456 400 266 349)(92 319 300 240 457 407 255 332)(93 326 289 223 458 414 244 345)(94 303 278 236 459 391 263 358)(95 310 297 219 460 398 252 341)(96 317 286 232 461 405 241 354)(97 324 275 215 462 412 260 337)(98 301 294 228 463 419 249 350)(99 308 283 211 464 396 268 333)(100 315 272 224 465 403 257 346)(101 322 291 237 466 410 246 359)(102 329 280 220 467 417 265 342)(103 306 299 233 468 394 254 355)(104 313 288 216 469 401 243 338)(105 320 277 229 470 408 262 351)(106 327 296 212 471 415 251 334)(107 304 285 225 472 392 270 347)(108 311 274 238 473 399 259 360)(109 318 293 221 474 406 248 343)(110 325 282 234 475 413 267 356)(111 302 271 217 476 420 256 339)(112 309 290 230 477 397 245 352)(113 316 279 213 478 404 264 335)(114 323 298 226 479 411 253 348)(115 330 287 239 480 418 242 331)(116 307 276 222 451 395 261 344)(117 314 295 235 452 402 250 357)(118 321 284 218 453 409 269 340)(119 328 273 231 454 416 258 353)(120 305 292 214 455 393 247 336)
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), (1,475,16,460)(2,474,17,459)(3,473,18,458)(4,472,19,457)(5,471,20,456)(6,470,21,455)(7,469,22,454)(8,468,23,453)(9,467,24,452)(10,466,25,451)(11,465,26,480)(12,464,27,479)(13,463,28,478)(14,462,29,477)(15,461,30,476)(31,407,46,392)(32,406,47,391)(33,405,48,420)(34,404,49,419)(35,403,50,418)(36,402,51,417)(37,401,52,416)(38,400,53,415)(39,399,54,414)(40,398,55,413)(41,397,56,412)(42,396,57,411)(43,395,58,410)(44,394,59,409)(45,393,60,408)(61,332,76,347)(62,331,77,346)(63,360,78,345)(64,359,79,344)(65,358,80,343)(66,357,81,342)(67,356,82,341)(68,355,83,340)(69,354,84,339)(70,353,85,338)(71,352,86,337)(72,351,87,336)(73,350,88,335)(74,349,89,334)(75,348,90,333)(91,188,106,203)(92,187,107,202)(93,186,108,201)(94,185,109,200)(95,184,110,199)(96,183,111,198)(97,182,112,197)(98,181,113,196)(99,210,114,195)(100,209,115,194)(101,208,116,193)(102,207,117,192)(103,206,118,191)(104,205,119,190)(105,204,120,189)(121,296,136,281)(122,295,137,280)(123,294,138,279)(124,293,139,278)(125,292,140,277)(126,291,141,276)(127,290,142,275)(128,289,143,274)(129,288,144,273)(130,287,145,272)(131,286,146,271)(132,285,147,300)(133,284,148,299)(134,283,149,298)(135,282,150,297)(151,266,166,251)(152,265,167,250)(153,264,168,249)(154,263,169,248)(155,262,170,247)(156,261,171,246)(157,260,172,245)(158,259,173,244)(159,258,174,243)(160,257,175,242)(161,256,176,241)(162,255,177,270)(163,254,178,269)(164,253,179,268)(165,252,180,267)(211,436,226,421)(212,435,227,450)(213,434,228,449)(214,433,229,448)(215,432,230,447)(216,431,231,446)(217,430,232,445)(218,429,233,444)(219,428,234,443)(220,427,235,442)(221,426,236,441)(222,425,237,440)(223,424,238,439)(224,423,239,438)(225,422,240,437)(301,383,316,368)(302,382,317,367)(303,381,318,366)(304,380,319,365)(305,379,320,364)(306,378,321,363)(307,377,322,362)(308,376,323,361)(309,375,324,390)(310,374,325,389)(311,373,326,388)(312,372,327,387)(313,371,328,386)(314,370,329,385)(315,369,330,384), (1,40,180,82,184,389,135,443)(2,47,169,65,185,366,124,426)(3,54,158,78,186,373,143,439)(4,31,177,61,187,380,132,422)(5,38,166,74,188,387,121,435)(6,45,155,87,189,364,140,448)(7,52,174,70,190,371,129,431)(8,59,163,83,191,378,148,444)(9,36,152,66,192,385,137,427)(10,43,171,79,193,362,126,440)(11,50,160,62,194,369,145,423)(12,57,179,75,195,376,134,436)(13,34,168,88,196,383,123,449)(14,41,157,71,197,390,142,432)(15,48,176,84,198,367,131,445)(16,55,165,67,199,374,150,428)(17,32,154,80,200,381,139,441)(18,39,173,63,201,388,128,424)(19,46,162,76,202,365,147,437)(20,53,151,89,203,372,136,450)(21,60,170,72,204,379,125,433)(22,37,159,85,205,386,144,446)(23,44,178,68,206,363,133,429)(24,51,167,81,207,370,122,442)(25,58,156,64,208,377,141,425)(26,35,175,77,209,384,130,438)(27,42,164,90,210,361,149,421)(28,49,153,73,181,368,138,434)(29,56,172,86,182,375,127,447)(30,33,161,69,183,382,146,430)(91,312,281,227,456,400,266,349)(92,319,300,240,457,407,255,332)(93,326,289,223,458,414,244,345)(94,303,278,236,459,391,263,358)(95,310,297,219,460,398,252,341)(96,317,286,232,461,405,241,354)(97,324,275,215,462,412,260,337)(98,301,294,228,463,419,249,350)(99,308,283,211,464,396,268,333)(100,315,272,224,465,403,257,346)(101,322,291,237,466,410,246,359)(102,329,280,220,467,417,265,342)(103,306,299,233,468,394,254,355)(104,313,288,216,469,401,243,338)(105,320,277,229,470,408,262,351)(106,327,296,212,471,415,251,334)(107,304,285,225,472,392,270,347)(108,311,274,238,473,399,259,360)(109,318,293,221,474,406,248,343)(110,325,282,234,475,413,267,356)(111,302,271,217,476,420,256,339)(112,309,290,230,477,397,245,352)(113,316,279,213,478,404,264,335)(114,323,298,226,479,411,253,348)(115,330,287,239,480,418,242,331)(116,307,276,222,451,395,261,344)(117,314,295,235,452,402,250,357)(118,321,284,218,453,409,269,340)(119,328,273,231,454,416,258,353)(120,305,292,214,455,393,247,336)>;
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), (1,475,16,460)(2,474,17,459)(3,473,18,458)(4,472,19,457)(5,471,20,456)(6,470,21,455)(7,469,22,454)(8,468,23,453)(9,467,24,452)(10,466,25,451)(11,465,26,480)(12,464,27,479)(13,463,28,478)(14,462,29,477)(15,461,30,476)(31,407,46,392)(32,406,47,391)(33,405,48,420)(34,404,49,419)(35,403,50,418)(36,402,51,417)(37,401,52,416)(38,400,53,415)(39,399,54,414)(40,398,55,413)(41,397,56,412)(42,396,57,411)(43,395,58,410)(44,394,59,409)(45,393,60,408)(61,332,76,347)(62,331,77,346)(63,360,78,345)(64,359,79,344)(65,358,80,343)(66,357,81,342)(67,356,82,341)(68,355,83,340)(69,354,84,339)(70,353,85,338)(71,352,86,337)(72,351,87,336)(73,350,88,335)(74,349,89,334)(75,348,90,333)(91,188,106,203)(92,187,107,202)(93,186,108,201)(94,185,109,200)(95,184,110,199)(96,183,111,198)(97,182,112,197)(98,181,113,196)(99,210,114,195)(100,209,115,194)(101,208,116,193)(102,207,117,192)(103,206,118,191)(104,205,119,190)(105,204,120,189)(121,296,136,281)(122,295,137,280)(123,294,138,279)(124,293,139,278)(125,292,140,277)(126,291,141,276)(127,290,142,275)(128,289,143,274)(129,288,144,273)(130,287,145,272)(131,286,146,271)(132,285,147,300)(133,284,148,299)(134,283,149,298)(135,282,150,297)(151,266,166,251)(152,265,167,250)(153,264,168,249)(154,263,169,248)(155,262,170,247)(156,261,171,246)(157,260,172,245)(158,259,173,244)(159,258,174,243)(160,257,175,242)(161,256,176,241)(162,255,177,270)(163,254,178,269)(164,253,179,268)(165,252,180,267)(211,436,226,421)(212,435,227,450)(213,434,228,449)(214,433,229,448)(215,432,230,447)(216,431,231,446)(217,430,232,445)(218,429,233,444)(219,428,234,443)(220,427,235,442)(221,426,236,441)(222,425,237,440)(223,424,238,439)(224,423,239,438)(225,422,240,437)(301,383,316,368)(302,382,317,367)(303,381,318,366)(304,380,319,365)(305,379,320,364)(306,378,321,363)(307,377,322,362)(308,376,323,361)(309,375,324,390)(310,374,325,389)(311,373,326,388)(312,372,327,387)(313,371,328,386)(314,370,329,385)(315,369,330,384), (1,40,180,82,184,389,135,443)(2,47,169,65,185,366,124,426)(3,54,158,78,186,373,143,439)(4,31,177,61,187,380,132,422)(5,38,166,74,188,387,121,435)(6,45,155,87,189,364,140,448)(7,52,174,70,190,371,129,431)(8,59,163,83,191,378,148,444)(9,36,152,66,192,385,137,427)(10,43,171,79,193,362,126,440)(11,50,160,62,194,369,145,423)(12,57,179,75,195,376,134,436)(13,34,168,88,196,383,123,449)(14,41,157,71,197,390,142,432)(15,48,176,84,198,367,131,445)(16,55,165,67,199,374,150,428)(17,32,154,80,200,381,139,441)(18,39,173,63,201,388,128,424)(19,46,162,76,202,365,147,437)(20,53,151,89,203,372,136,450)(21,60,170,72,204,379,125,433)(22,37,159,85,205,386,144,446)(23,44,178,68,206,363,133,429)(24,51,167,81,207,370,122,442)(25,58,156,64,208,377,141,425)(26,35,175,77,209,384,130,438)(27,42,164,90,210,361,149,421)(28,49,153,73,181,368,138,434)(29,56,172,86,182,375,127,447)(30,33,161,69,183,382,146,430)(91,312,281,227,456,400,266,349)(92,319,300,240,457,407,255,332)(93,326,289,223,458,414,244,345)(94,303,278,236,459,391,263,358)(95,310,297,219,460,398,252,341)(96,317,286,232,461,405,241,354)(97,324,275,215,462,412,260,337)(98,301,294,228,463,419,249,350)(99,308,283,211,464,396,268,333)(100,315,272,224,465,403,257,346)(101,322,291,237,466,410,246,359)(102,329,280,220,467,417,265,342)(103,306,299,233,468,394,254,355)(104,313,288,216,469,401,243,338)(105,320,277,229,470,408,262,351)(106,327,296,212,471,415,251,334)(107,304,285,225,472,392,270,347)(108,311,274,238,473,399,259,360)(109,318,293,221,474,406,248,343)(110,325,282,234,475,413,267,356)(111,302,271,217,476,420,256,339)(112,309,290,230,477,397,245,352)(113,316,279,213,478,404,264,335)(114,323,298,226,479,411,253,348)(115,330,287,239,480,418,242,331)(116,307,276,222,451,395,261,344)(117,314,295,235,452,402,250,357)(118,321,284,218,453,409,269,340)(119,328,273,231,454,416,258,353)(120,305,292,214,455,393,247,336) );
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)], [(1,475,16,460),(2,474,17,459),(3,473,18,458),(4,472,19,457),(5,471,20,456),(6,470,21,455),(7,469,22,454),(8,468,23,453),(9,467,24,452),(10,466,25,451),(11,465,26,480),(12,464,27,479),(13,463,28,478),(14,462,29,477),(15,461,30,476),(31,407,46,392),(32,406,47,391),(33,405,48,420),(34,404,49,419),(35,403,50,418),(36,402,51,417),(37,401,52,416),(38,400,53,415),(39,399,54,414),(40,398,55,413),(41,397,56,412),(42,396,57,411),(43,395,58,410),(44,394,59,409),(45,393,60,408),(61,332,76,347),(62,331,77,346),(63,360,78,345),(64,359,79,344),(65,358,80,343),(66,357,81,342),(67,356,82,341),(68,355,83,340),(69,354,84,339),(70,353,85,338),(71,352,86,337),(72,351,87,336),(73,350,88,335),(74,349,89,334),(75,348,90,333),(91,188,106,203),(92,187,107,202),(93,186,108,201),(94,185,109,200),(95,184,110,199),(96,183,111,198),(97,182,112,197),(98,181,113,196),(99,210,114,195),(100,209,115,194),(101,208,116,193),(102,207,117,192),(103,206,118,191),(104,205,119,190),(105,204,120,189),(121,296,136,281),(122,295,137,280),(123,294,138,279),(124,293,139,278),(125,292,140,277),(126,291,141,276),(127,290,142,275),(128,289,143,274),(129,288,144,273),(130,287,145,272),(131,286,146,271),(132,285,147,300),(133,284,148,299),(134,283,149,298),(135,282,150,297),(151,266,166,251),(152,265,167,250),(153,264,168,249),(154,263,169,248),(155,262,170,247),(156,261,171,246),(157,260,172,245),(158,259,173,244),(159,258,174,243),(160,257,175,242),(161,256,176,241),(162,255,177,270),(163,254,178,269),(164,253,179,268),(165,252,180,267),(211,436,226,421),(212,435,227,450),(213,434,228,449),(214,433,229,448),(215,432,230,447),(216,431,231,446),(217,430,232,445),(218,429,233,444),(219,428,234,443),(220,427,235,442),(221,426,236,441),(222,425,237,440),(223,424,238,439),(224,423,239,438),(225,422,240,437),(301,383,316,368),(302,382,317,367),(303,381,318,366),(304,380,319,365),(305,379,320,364),(306,378,321,363),(307,377,322,362),(308,376,323,361),(309,375,324,390),(310,374,325,389),(311,373,326,388),(312,372,327,387),(313,371,328,386),(314,370,329,385),(315,369,330,384)], [(1,40,180,82,184,389,135,443),(2,47,169,65,185,366,124,426),(3,54,158,78,186,373,143,439),(4,31,177,61,187,380,132,422),(5,38,166,74,188,387,121,435),(6,45,155,87,189,364,140,448),(7,52,174,70,190,371,129,431),(8,59,163,83,191,378,148,444),(9,36,152,66,192,385,137,427),(10,43,171,79,193,362,126,440),(11,50,160,62,194,369,145,423),(12,57,179,75,195,376,134,436),(13,34,168,88,196,383,123,449),(14,41,157,71,197,390,142,432),(15,48,176,84,198,367,131,445),(16,55,165,67,199,374,150,428),(17,32,154,80,200,381,139,441),(18,39,173,63,201,388,128,424),(19,46,162,76,202,365,147,437),(20,53,151,89,203,372,136,450),(21,60,170,72,204,379,125,433),(22,37,159,85,205,386,144,446),(23,44,178,68,206,363,133,429),(24,51,167,81,207,370,122,442),(25,58,156,64,208,377,141,425),(26,35,175,77,209,384,130,438),(27,42,164,90,210,361,149,421),(28,49,153,73,181,368,138,434),(29,56,172,86,182,375,127,447),(30,33,161,69,183,382,146,430),(91,312,281,227,456,400,266,349),(92,319,300,240,457,407,255,332),(93,326,289,223,458,414,244,345),(94,303,278,236,459,391,263,358),(95,310,297,219,460,398,252,341),(96,317,286,232,461,405,241,354),(97,324,275,215,462,412,260,337),(98,301,294,228,463,419,249,350),(99,308,283,211,464,396,268,333),(100,315,272,224,465,403,257,346),(101,322,291,237,466,410,246,359),(102,329,280,220,467,417,265,342),(103,306,299,233,468,394,254,355),(104,313,288,216,469,401,243,338),(105,320,277,229,470,408,262,351),(106,327,296,212,471,415,251,334),(107,304,285,225,472,392,270,347),(108,311,274,238,473,399,259,360),(109,318,293,221,474,406,248,343),(110,325,282,234,475,413,267,356),(111,302,271,217,476,420,256,339),(112,309,290,230,477,397,245,352),(113,316,279,213,478,404,264,335),(114,323,298,226,479,411,253,348),(115,330,287,239,480,418,242,331),(116,307,276,222,451,395,261,344),(117,314,295,235,452,402,250,357),(118,321,284,218,453,409,269,340),(119,328,273,231,454,416,258,353),(120,305,292,214,455,393,247,336)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15 | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | 30 | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 30 | 30 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 8 | 12 | 12 | 12 | 12 | 10 | ··· | 10 | 8 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | + | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | Q8 | D6 | M4(2) | Dic6 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 | F5 | C2×F5 | D5⋊C8 | C4⋊F5 | C22.F5 | S3×F5 | Dic3⋊F5 | D15⋊C8 | D6.F5 |
| kernel | Dic15⋊C8 | Dic3×Dic5 | C6×C5⋊C8 | C2×C15⋊C8 | C10×Dic3 | C2×Dic15 | Dic15 | C2×C5⋊C8 | C3×Dic5 | C3×Dic5 | C2×Dic5 | C30 | Dic5 | Dic5 | C2×C10 | C10 | C10 | C2×Dic3 | C2×C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 |
Matrix representation of Dic15⋊C8 ►in GL8(𝔽241)
| 240 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 239 | 184 | 115 | 133 |
| 0 | 0 | 0 | 0 | 182 | 58 | 7 | 185 |
| 0 | 0 | 0 | 0 | 56 | 191 | 59 | 183 |
| 0 | 0 | 0 | 0 | 189 | 2 | 57 | 126 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 76 | 238 | 0 | 0 | 0 | 0 |
| 0 | 0 | 238 | 165 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 228 | 237 | 97 | 236 |
| 0 | 0 | 0 | 0 | 84 | 232 | 22 | 223 |
| 0 | 0 | 0 | 0 | 9 | 219 | 18 | 79 |
| 0 | 0 | 0 | 0 | 5 | 75 | 13 | 4 |
G:=sub<GL(8,GF(241))| [240,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0],[1,0,0,0,0,0,0,0,240,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,239,182,56,189,0,0,0,0,184,58,191,2,0,0,0,0,115,7,59,57,0,0,0,0,133,185,183,126],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,76,238,0,0,0,0,0,0,238,165,0,0,0,0,0,0,0,0,228,84,9,5,0,0,0,0,237,232,219,75,0,0,0,0,97,22,18,13,0,0,0,0,236,223,79,4] >;
Dic15⋊C8 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes C_8 % in TeX
G:=Group("Dic15:C8"); // GroupNames label
G:=SmallGroup(480,253);
// by ID
G=gap.SmallGroup(480,253);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,176,100,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c|a^30=c^8=1,b^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^15*b>;
// generators/relations