metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15⋊10Q8, Dic30⋊11C4, C15⋊18(C4×Q8), C4⋊C4.7D15, C4.4(C4×D15), C6.38(Q8×D5), C2.1(Q8×D15), C20.59(C4×S3), C60.84(C2×C4), C12.27(C4×D5), (C2×C4).28D30, C30.91(C2×Q8), C10.38(S3×Q8), (C2×C20).208D6, (C2×C12).206D10, C3⋊4(Dic5⋊3Q8), C5⋊5(Dic6⋊C4), (C2×Dic30).8C2, C30.4Q8.5C2, C30.218(C4○D4), C6.95(D4⋊2D5), C2.3(D4⋊2D15), C30.159(C22×C4), (C2×C60).176C22, (C2×C30).284C23, Dic15.30(C2×C4), (C4×Dic15).10C2, C10.95(D4⋊2S3), C22.15(C22×D15), (C2×Dic15).160C22, C6.64(C2×C4×D5), (C5×C4⋊C4).4S3, C10.96(S3×C2×C4), (C3×C4⋊C4).4D5, C2.10(C2×C4×D15), (C15×C4⋊C4).4C2, (C2×C6).280(C22×D5), (C2×C10).279(C22×S3), SmallGroup(480,852)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15⋊10Q8
G = < a,b,c | a60=c4=1, b2=a30, bab-1=a-1, cac-1=a31, bc=cb >
Subgroups: 612 in 140 conjugacy classes, 65 normal (33 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, C2×C4, Q8, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C4⋊C4, C2×Q8, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×C12, C2×C12, C30, C4×Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, Dic15, Dic15, C60, C60, C2×C30, C4×Dic5, C10.D4, C5×C4⋊C4, C2×Dic10, Dic6⋊C4, Dic30, C2×Dic15, C2×Dic15, C2×C60, C2×C60, Dic5⋊3Q8, C4×Dic15, C4×Dic15, C30.4Q8, C15×C4⋊C4, C2×Dic30, Dic15⋊10Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D5, D6, C22×C4, C2×Q8, C4○D4, D10, C4×S3, C22×S3, D15, C4×Q8, C4×D5, C22×D5, S3×C2×C4, D4⋊2S3, S3×Q8, D30, C2×C4×D5, D4⋊2D5, Q8×D5, Dic6⋊C4, C4×D15, C22×D15, Dic5⋊3Q8, C2×C4×D15, D4⋊2D15, Q8×D15, Dic15⋊10Q8
►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)
(1 470 31 440)(2 469 32 439)(3 468 33 438)(4 467 34 437)(5 466 35 436)(6 465 36 435)(7 464 37 434)(8 463 38 433)(9 462 39 432)(10 461 40 431)(11 460 41 430)(12 459 42 429)(13 458 43 428)(14 457 44 427)(15 456 45 426)(16 455 46 425)(17 454 47 424)(18 453 48 423)(19 452 49 422)(20 451 50 421)(21 450 51 480)(22 449 52 479)(23 448 53 478)(24 447 54 477)(25 446 55 476)(26 445 56 475)(27 444 57 474)(28 443 58 473)(29 442 59 472)(30 441 60 471)(61 397 91 367)(62 396 92 366)(63 395 93 365)(64 394 94 364)(65 393 95 363)(66 392 96 362)(67 391 97 361)(68 390 98 420)(69 389 99 419)(70 388 100 418)(71 387 101 417)(72 386 102 416)(73 385 103 415)(74 384 104 414)(75 383 105 413)(76 382 106 412)(77 381 107 411)(78 380 108 410)(79 379 109 409)(80 378 110 408)(81 377 111 407)(82 376 112 406)(83 375 113 405)(84 374 114 404)(85 373 115 403)(86 372 116 402)(87 371 117 401)(88 370 118 400)(89 369 119 399)(90 368 120 398)(121 192 151 222)(122 191 152 221)(123 190 153 220)(124 189 154 219)(125 188 155 218)(126 187 156 217)(127 186 157 216)(128 185 158 215)(129 184 159 214)(130 183 160 213)(131 182 161 212)(132 181 162 211)(133 240 163 210)(134 239 164 209)(135 238 165 208)(136 237 166 207)(137 236 167 206)(138 235 168 205)(139 234 169 204)(140 233 170 203)(141 232 171 202)(142 231 172 201)(143 230 173 200)(144 229 174 199)(145 228 175 198)(146 227 176 197)(147 226 177 196)(148 225 178 195)(149 224 179 194)(150 223 180 193)(241 338 271 308)(242 337 272 307)(243 336 273 306)(244 335 274 305)(245 334 275 304)(246 333 276 303)(247 332 277 302)(248 331 278 301)(249 330 279 360)(250 329 280 359)(251 328 281 358)(252 327 282 357)(253 326 283 356)(254 325 284 355)(255 324 285 354)(256 323 286 353)(257 322 287 352)(258 321 288 351)(259 320 289 350)(260 319 290 349)(261 318 291 348)(262 317 292 347)(263 316 293 346)(264 315 294 345)(265 314 295 344)(266 313 296 343)(267 312 297 342)(268 311 298 341)(269 310 299 340)(270 309 300 339)
(1 288 188 367)(2 259 189 398)(3 290 190 369)(4 261 191 400)(5 292 192 371)(6 263 193 402)(7 294 194 373)(8 265 195 404)(9 296 196 375)(10 267 197 406)(11 298 198 377)(12 269 199 408)(13 300 200 379)(14 271 201 410)(15 242 202 381)(16 273 203 412)(17 244 204 383)(18 275 205 414)(19 246 206 385)(20 277 207 416)(21 248 208 387)(22 279 209 418)(23 250 210 389)(24 281 211 420)(25 252 212 391)(26 283 213 362)(27 254 214 393)(28 285 215 364)(29 256 216 395)(30 287 217 366)(31 258 218 397)(32 289 219 368)(33 260 220 399)(34 291 221 370)(35 262 222 401)(36 293 223 372)(37 264 224 403)(38 295 225 374)(39 266 226 405)(40 297 227 376)(41 268 228 407)(42 299 229 378)(43 270 230 409)(44 241 231 380)(45 272 232 411)(46 243 233 382)(47 274 234 413)(48 245 235 384)(49 276 236 415)(50 247 237 386)(51 278 238 417)(52 249 239 388)(53 280 240 419)(54 251 181 390)(55 282 182 361)(56 253 183 392)(57 284 184 363)(58 255 185 394)(59 286 186 365)(60 257 187 396)(61 470 351 155)(62 441 352 126)(63 472 353 157)(64 443 354 128)(65 474 355 159)(66 445 356 130)(67 476 357 161)(68 447 358 132)(69 478 359 163)(70 449 360 134)(71 480 301 165)(72 451 302 136)(73 422 303 167)(74 453 304 138)(75 424 305 169)(76 455 306 140)(77 426 307 171)(78 457 308 142)(79 428 309 173)(80 459 310 144)(81 430 311 175)(82 461 312 146)(83 432 313 177)(84 463 314 148)(85 434 315 179)(86 465 316 150)(87 436 317 121)(88 467 318 152)(89 438 319 123)(90 469 320 154)(91 440 321 125)(92 471 322 156)(93 442 323 127)(94 473 324 158)(95 444 325 129)(96 475 326 160)(97 446 327 131)(98 477 328 162)(99 448 329 133)(100 479 330 164)(101 450 331 135)(102 421 332 166)(103 452 333 137)(104 423 334 168)(105 454 335 139)(106 425 336 170)(107 456 337 141)(108 427 338 172)(109 458 339 143)(110 429 340 174)(111 460 341 145)(112 431 342 176)(113 462 343 147)(114 433 344 178)(115 464 345 149)(116 435 346 180)(117 466 347 151)(118 437 348 122)(119 468 349 153)(120 439 350 124)
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), (1,470,31,440)(2,469,32,439)(3,468,33,438)(4,467,34,437)(5,466,35,436)(6,465,36,435)(7,464,37,434)(8,463,38,433)(9,462,39,432)(10,461,40,431)(11,460,41,430)(12,459,42,429)(13,458,43,428)(14,457,44,427)(15,456,45,426)(16,455,46,425)(17,454,47,424)(18,453,48,423)(19,452,49,422)(20,451,50,421)(21,450,51,480)(22,449,52,479)(23,448,53,478)(24,447,54,477)(25,446,55,476)(26,445,56,475)(27,444,57,474)(28,443,58,473)(29,442,59,472)(30,441,60,471)(61,397,91,367)(62,396,92,366)(63,395,93,365)(64,394,94,364)(65,393,95,363)(66,392,96,362)(67,391,97,361)(68,390,98,420)(69,389,99,419)(70,388,100,418)(71,387,101,417)(72,386,102,416)(73,385,103,415)(74,384,104,414)(75,383,105,413)(76,382,106,412)(77,381,107,411)(78,380,108,410)(79,379,109,409)(80,378,110,408)(81,377,111,407)(82,376,112,406)(83,375,113,405)(84,374,114,404)(85,373,115,403)(86,372,116,402)(87,371,117,401)(88,370,118,400)(89,369,119,399)(90,368,120,398)(121,192,151,222)(122,191,152,221)(123,190,153,220)(124,189,154,219)(125,188,155,218)(126,187,156,217)(127,186,157,216)(128,185,158,215)(129,184,159,214)(130,183,160,213)(131,182,161,212)(132,181,162,211)(133,240,163,210)(134,239,164,209)(135,238,165,208)(136,237,166,207)(137,236,167,206)(138,235,168,205)(139,234,169,204)(140,233,170,203)(141,232,171,202)(142,231,172,201)(143,230,173,200)(144,229,174,199)(145,228,175,198)(146,227,176,197)(147,226,177,196)(148,225,178,195)(149,224,179,194)(150,223,180,193)(241,338,271,308)(242,337,272,307)(243,336,273,306)(244,335,274,305)(245,334,275,304)(246,333,276,303)(247,332,277,302)(248,331,278,301)(249,330,279,360)(250,329,280,359)(251,328,281,358)(252,327,282,357)(253,326,283,356)(254,325,284,355)(255,324,285,354)(256,323,286,353)(257,322,287,352)(258,321,288,351)(259,320,289,350)(260,319,290,349)(261,318,291,348)(262,317,292,347)(263,316,293,346)(264,315,294,345)(265,314,295,344)(266,313,296,343)(267,312,297,342)(268,311,298,341)(269,310,299,340)(270,309,300,339), (1,288,188,367)(2,259,189,398)(3,290,190,369)(4,261,191,400)(5,292,192,371)(6,263,193,402)(7,294,194,373)(8,265,195,404)(9,296,196,375)(10,267,197,406)(11,298,198,377)(12,269,199,408)(13,300,200,379)(14,271,201,410)(15,242,202,381)(16,273,203,412)(17,244,204,383)(18,275,205,414)(19,246,206,385)(20,277,207,416)(21,248,208,387)(22,279,209,418)(23,250,210,389)(24,281,211,420)(25,252,212,391)(26,283,213,362)(27,254,214,393)(28,285,215,364)(29,256,216,395)(30,287,217,366)(31,258,218,397)(32,289,219,368)(33,260,220,399)(34,291,221,370)(35,262,222,401)(36,293,223,372)(37,264,224,403)(38,295,225,374)(39,266,226,405)(40,297,227,376)(41,268,228,407)(42,299,229,378)(43,270,230,409)(44,241,231,380)(45,272,232,411)(46,243,233,382)(47,274,234,413)(48,245,235,384)(49,276,236,415)(50,247,237,386)(51,278,238,417)(52,249,239,388)(53,280,240,419)(54,251,181,390)(55,282,182,361)(56,253,183,392)(57,284,184,363)(58,255,185,394)(59,286,186,365)(60,257,187,396)(61,470,351,155)(62,441,352,126)(63,472,353,157)(64,443,354,128)(65,474,355,159)(66,445,356,130)(67,476,357,161)(68,447,358,132)(69,478,359,163)(70,449,360,134)(71,480,301,165)(72,451,302,136)(73,422,303,167)(74,453,304,138)(75,424,305,169)(76,455,306,140)(77,426,307,171)(78,457,308,142)(79,428,309,173)(80,459,310,144)(81,430,311,175)(82,461,312,146)(83,432,313,177)(84,463,314,148)(85,434,315,179)(86,465,316,150)(87,436,317,121)(88,467,318,152)(89,438,319,123)(90,469,320,154)(91,440,321,125)(92,471,322,156)(93,442,323,127)(94,473,324,158)(95,444,325,129)(96,475,326,160)(97,446,327,131)(98,477,328,162)(99,448,329,133)(100,479,330,164)(101,450,331,135)(102,421,332,166)(103,452,333,137)(104,423,334,168)(105,454,335,139)(106,425,336,170)(107,456,337,141)(108,427,338,172)(109,458,339,143)(110,429,340,174)(111,460,341,145)(112,431,342,176)(113,462,343,147)(114,433,344,178)(115,464,345,149)(116,435,346,180)(117,466,347,151)(118,437,348,122)(119,468,349,153)(120,439,350,124)>;
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), (1,470,31,440)(2,469,32,439)(3,468,33,438)(4,467,34,437)(5,466,35,436)(6,465,36,435)(7,464,37,434)(8,463,38,433)(9,462,39,432)(10,461,40,431)(11,460,41,430)(12,459,42,429)(13,458,43,428)(14,457,44,427)(15,456,45,426)(16,455,46,425)(17,454,47,424)(18,453,48,423)(19,452,49,422)(20,451,50,421)(21,450,51,480)(22,449,52,479)(23,448,53,478)(24,447,54,477)(25,446,55,476)(26,445,56,475)(27,444,57,474)(28,443,58,473)(29,442,59,472)(30,441,60,471)(61,397,91,367)(62,396,92,366)(63,395,93,365)(64,394,94,364)(65,393,95,363)(66,392,96,362)(67,391,97,361)(68,390,98,420)(69,389,99,419)(70,388,100,418)(71,387,101,417)(72,386,102,416)(73,385,103,415)(74,384,104,414)(75,383,105,413)(76,382,106,412)(77,381,107,411)(78,380,108,410)(79,379,109,409)(80,378,110,408)(81,377,111,407)(82,376,112,406)(83,375,113,405)(84,374,114,404)(85,373,115,403)(86,372,116,402)(87,371,117,401)(88,370,118,400)(89,369,119,399)(90,368,120,398)(121,192,151,222)(122,191,152,221)(123,190,153,220)(124,189,154,219)(125,188,155,218)(126,187,156,217)(127,186,157,216)(128,185,158,215)(129,184,159,214)(130,183,160,213)(131,182,161,212)(132,181,162,211)(133,240,163,210)(134,239,164,209)(135,238,165,208)(136,237,166,207)(137,236,167,206)(138,235,168,205)(139,234,169,204)(140,233,170,203)(141,232,171,202)(142,231,172,201)(143,230,173,200)(144,229,174,199)(145,228,175,198)(146,227,176,197)(147,226,177,196)(148,225,178,195)(149,224,179,194)(150,223,180,193)(241,338,271,308)(242,337,272,307)(243,336,273,306)(244,335,274,305)(245,334,275,304)(246,333,276,303)(247,332,277,302)(248,331,278,301)(249,330,279,360)(250,329,280,359)(251,328,281,358)(252,327,282,357)(253,326,283,356)(254,325,284,355)(255,324,285,354)(256,323,286,353)(257,322,287,352)(258,321,288,351)(259,320,289,350)(260,319,290,349)(261,318,291,348)(262,317,292,347)(263,316,293,346)(264,315,294,345)(265,314,295,344)(266,313,296,343)(267,312,297,342)(268,311,298,341)(269,310,299,340)(270,309,300,339), (1,288,188,367)(2,259,189,398)(3,290,190,369)(4,261,191,400)(5,292,192,371)(6,263,193,402)(7,294,194,373)(8,265,195,404)(9,296,196,375)(10,267,197,406)(11,298,198,377)(12,269,199,408)(13,300,200,379)(14,271,201,410)(15,242,202,381)(16,273,203,412)(17,244,204,383)(18,275,205,414)(19,246,206,385)(20,277,207,416)(21,248,208,387)(22,279,209,418)(23,250,210,389)(24,281,211,420)(25,252,212,391)(26,283,213,362)(27,254,214,393)(28,285,215,364)(29,256,216,395)(30,287,217,366)(31,258,218,397)(32,289,219,368)(33,260,220,399)(34,291,221,370)(35,262,222,401)(36,293,223,372)(37,264,224,403)(38,295,225,374)(39,266,226,405)(40,297,227,376)(41,268,228,407)(42,299,229,378)(43,270,230,409)(44,241,231,380)(45,272,232,411)(46,243,233,382)(47,274,234,413)(48,245,235,384)(49,276,236,415)(50,247,237,386)(51,278,238,417)(52,249,239,388)(53,280,240,419)(54,251,181,390)(55,282,182,361)(56,253,183,392)(57,284,184,363)(58,255,185,394)(59,286,186,365)(60,257,187,396)(61,470,351,155)(62,441,352,126)(63,472,353,157)(64,443,354,128)(65,474,355,159)(66,445,356,130)(67,476,357,161)(68,447,358,132)(69,478,359,163)(70,449,360,134)(71,480,301,165)(72,451,302,136)(73,422,303,167)(74,453,304,138)(75,424,305,169)(76,455,306,140)(77,426,307,171)(78,457,308,142)(79,428,309,173)(80,459,310,144)(81,430,311,175)(82,461,312,146)(83,432,313,177)(84,463,314,148)(85,434,315,179)(86,465,316,150)(87,436,317,121)(88,467,318,152)(89,438,319,123)(90,469,320,154)(91,440,321,125)(92,471,322,156)(93,442,323,127)(94,473,324,158)(95,444,325,129)(96,475,326,160)(97,446,327,131)(98,477,328,162)(99,448,329,133)(100,479,330,164)(101,450,331,135)(102,421,332,166)(103,452,333,137)(104,423,334,168)(105,454,335,139)(106,425,336,170)(107,456,337,141)(108,427,338,172)(109,458,339,143)(110,429,340,174)(111,460,341,145)(112,431,342,176)(113,462,343,147)(114,433,344,178)(115,464,345,149)(116,435,346,180)(117,466,347,151)(118,437,348,122)(119,468,349,153)(120,439,350,124) );
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)], [(1,470,31,440),(2,469,32,439),(3,468,33,438),(4,467,34,437),(5,466,35,436),(6,465,36,435),(7,464,37,434),(8,463,38,433),(9,462,39,432),(10,461,40,431),(11,460,41,430),(12,459,42,429),(13,458,43,428),(14,457,44,427),(15,456,45,426),(16,455,46,425),(17,454,47,424),(18,453,48,423),(19,452,49,422),(20,451,50,421),(21,450,51,480),(22,449,52,479),(23,448,53,478),(24,447,54,477),(25,446,55,476),(26,445,56,475),(27,444,57,474),(28,443,58,473),(29,442,59,472),(30,441,60,471),(61,397,91,367),(62,396,92,366),(63,395,93,365),(64,394,94,364),(65,393,95,363),(66,392,96,362),(67,391,97,361),(68,390,98,420),(69,389,99,419),(70,388,100,418),(71,387,101,417),(72,386,102,416),(73,385,103,415),(74,384,104,414),(75,383,105,413),(76,382,106,412),(77,381,107,411),(78,380,108,410),(79,379,109,409),(80,378,110,408),(81,377,111,407),(82,376,112,406),(83,375,113,405),(84,374,114,404),(85,373,115,403),(86,372,116,402),(87,371,117,401),(88,370,118,400),(89,369,119,399),(90,368,120,398),(121,192,151,222),(122,191,152,221),(123,190,153,220),(124,189,154,219),(125,188,155,218),(126,187,156,217),(127,186,157,216),(128,185,158,215),(129,184,159,214),(130,183,160,213),(131,182,161,212),(132,181,162,211),(133,240,163,210),(134,239,164,209),(135,238,165,208),(136,237,166,207),(137,236,167,206),(138,235,168,205),(139,234,169,204),(140,233,170,203),(141,232,171,202),(142,231,172,201),(143,230,173,200),(144,229,174,199),(145,228,175,198),(146,227,176,197),(147,226,177,196),(148,225,178,195),(149,224,179,194),(150,223,180,193),(241,338,271,308),(242,337,272,307),(243,336,273,306),(244,335,274,305),(245,334,275,304),(246,333,276,303),(247,332,277,302),(248,331,278,301),(249,330,279,360),(250,329,280,359),(251,328,281,358),(252,327,282,357),(253,326,283,356),(254,325,284,355),(255,324,285,354),(256,323,286,353),(257,322,287,352),(258,321,288,351),(259,320,289,350),(260,319,290,349),(261,318,291,348),(262,317,292,347),(263,316,293,346),(264,315,294,345),(265,314,295,344),(266,313,296,343),(267,312,297,342),(268,311,298,341),(269,310,299,340),(270,309,300,339)], [(1,288,188,367),(2,259,189,398),(3,290,190,369),(4,261,191,400),(5,292,192,371),(6,263,193,402),(7,294,194,373),(8,265,195,404),(9,296,196,375),(10,267,197,406),(11,298,198,377),(12,269,199,408),(13,300,200,379),(14,271,201,410),(15,242,202,381),(16,273,203,412),(17,244,204,383),(18,275,205,414),(19,246,206,385),(20,277,207,416),(21,248,208,387),(22,279,209,418),(23,250,210,389),(24,281,211,420),(25,252,212,391),(26,283,213,362),(27,254,214,393),(28,285,215,364),(29,256,216,395),(30,287,217,366),(31,258,218,397),(32,289,219,368),(33,260,220,399),(34,291,221,370),(35,262,222,401),(36,293,223,372),(37,264,224,403),(38,295,225,374),(39,266,226,405),(40,297,227,376),(41,268,228,407),(42,299,229,378),(43,270,230,409),(44,241,231,380),(45,272,232,411),(46,243,233,382),(47,274,234,413),(48,245,235,384),(49,276,236,415),(50,247,237,386),(51,278,238,417),(52,249,239,388),(53,280,240,419),(54,251,181,390),(55,282,182,361),(56,253,183,392),(57,284,184,363),(58,255,185,394),(59,286,186,365),(60,257,187,396),(61,470,351,155),(62,441,352,126),(63,472,353,157),(64,443,354,128),(65,474,355,159),(66,445,356,130),(67,476,357,161),(68,447,358,132),(69,478,359,163),(70,449,360,134),(71,480,301,165),(72,451,302,136),(73,422,303,167),(74,453,304,138),(75,424,305,169),(76,455,306,140),(77,426,307,171),(78,457,308,142),(79,428,309,173),(80,459,310,144),(81,430,311,175),(82,461,312,146),(83,432,313,177),(84,463,314,148),(85,434,315,179),(86,465,316,150),(87,436,317,121),(88,467,318,152),(89,438,319,123),(90,469,320,154),(91,440,321,125),(92,471,322,156),(93,442,323,127),(94,473,324,158),(95,444,325,129),(96,475,326,160),(97,446,327,131),(98,477,328,162),(99,448,329,133),(100,479,330,164),(101,450,331,135),(102,421,332,166),(103,452,333,137),(104,423,334,168),(105,454,335,139),(106,425,336,170),(107,456,337,141),(108,427,338,172),(109,458,339,143),(110,429,340,174),(111,460,341,145),(112,431,342,176),(113,462,343,147),(114,433,344,178),(115,464,345,149),(116,435,346,180),(117,466,347,151),(118,437,348,122),(119,468,349,153),(120,439,350,124)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 15 | 15 | 15 | 15 | 30 | ··· | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - | - | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D5 | D6 | C4○D4 | D10 | C4×S3 | D15 | C4×D5 | D30 | C4×D15 | D4⋊2S3 | S3×Q8 | D4⋊2D5 | Q8×D5 | D4⋊2D15 | Q8×D15 |
| kernel | Dic15⋊10Q8 | C4×Dic15 | C30.4Q8 | C15×C4⋊C4 | C2×Dic30 | Dic30 | C5×C4⋊C4 | Dic15 | C3×C4⋊C4 | C2×C20 | C30 | C2×C12 | C20 | C4⋊C4 | C12 | C2×C4 | C4 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 3 | 2 | 1 | 1 | 8 | 1 | 2 | 2 | 3 | 2 | 6 | 4 | 4 | 8 | 12 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic15⋊10Q8 ►in GL5(𝔽61)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 37 | 0 | 0 |
| 0 | 24 | 47 | 0 | 0 |
| 0 | 0 | 0 | 27 | 32 |
| 0 | 0 | 0 | 42 | 34 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 44 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 60 | 21 |
| 0 | 0 | 0 | 58 | 1 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 | 13 |
| 0 | 0 | 0 | 33 | 50 |
G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,28,24,0,0,0,37,47,0,0,0,0,0,27,42,0,0,0,32,34],[1,0,0,0,0,0,1,0,0,0,0,44,60,0,0,0,0,0,60,58,0,0,0,21,1],[11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,33,0,0,0,13,50] >;
Dic15⋊10Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_{10}Q_8 % in TeX
G:=Group("Dic15:10Q8"); // GroupNames label
G:=SmallGroup(480,852);
// by ID
G=gap.SmallGroup(480,852);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,135,142,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=c^4=1,b^2=a^30,b*a*b^-1=a^-1,c*a*c^-1=a^31,b*c=c*b>;
// generators/relations