metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic15⋊4C8, C30.20C42, C15⋊10(C4×C8), C3⋊C8⋊5Dic5, C6.3(C8×D5), C5⋊4(C8×Dic3), C3⋊1(C8×Dic5), C10.12(S3×C8), C30.27(C2×C8), C12.76(C4×D5), C5⋊2C8⋊9Dic3, C6.3(C4×Dic5), C20.107(C4×S3), C60.143(C2×C4), (C2×C20).317D6, C4.19(S3×Dic5), C4.19(D5×Dic3), (C2×C12).321D10, C2.3(Dic3×Dic5), C10.15(C4×Dic3), C12.24(C2×Dic5), C20.45(C2×Dic3), C2.2(D15⋊2C8), (C2×C60).219C22, (C4×Dic15).17C2, (C2×Dic15).17C4, C22.8(D30.C2), (C5×C3⋊C8)⋊10C4, (C3×C5⋊2C8)⋊4C4, (C10×C3⋊C8).8C2, (C2×C3⋊C8).12D5, (C6×C5⋊2C8).8C2, (C2×C6).16(C4×D5), (C2×C5⋊2C8).14S3, (C2×C10).39(C4×S3), (C2×C30).73(C2×C4), (C2×C4).222(S3×D5), SmallGroup(480,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — Dic15⋊4C8 |
Generators and relations for Dic15⋊4C8
G = < a,b,c | a30=c8=1, b2=a15, bab-1=a-1, cac-1=a11, bc=cb >
Subgroups: 284 in 88 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, Dic3, C12, C2×C6, C15, C42, C2×C8, Dic5, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C30, C4×C8, C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, Dic15, C60, C2×C30, C2×C5⋊2C8, C4×Dic5, C2×C40, C8×Dic3, C5×C3⋊C8, C3×C5⋊2C8, C2×Dic15, C2×C60, C8×Dic5, C6×C5⋊2C8, C10×C3⋊C8, C4×Dic15, Dic15⋊4C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D5, Dic3, D6, C42, C2×C8, Dic5, D10, C4×S3, C2×Dic3, C4×C8, C4×D5, C2×Dic5, S3×C8, C4×Dic3, S3×D5, C8×D5, C4×Dic5, C8×Dic3, D5×Dic3, S3×Dic5, D30.C2, C8×Dic5, D15⋊2C8, Dic3×Dic5, Dic15⋊4C8
►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)
(1 193 16 208)(2 192 17 207)(3 191 18 206)(4 190 19 205)(5 189 20 204)(6 188 21 203)(7 187 22 202)(8 186 23 201)(9 185 24 200)(10 184 25 199)(11 183 26 198)(12 182 27 197)(13 181 28 196)(14 210 29 195)(15 209 30 194)(31 89 46 74)(32 88 47 73)(33 87 48 72)(34 86 49 71)(35 85 50 70)(36 84 51 69)(37 83 52 68)(38 82 53 67)(39 81 54 66)(40 80 55 65)(41 79 56 64)(42 78 57 63)(43 77 58 62)(44 76 59 61)(45 75 60 90)(91 236 106 221)(92 235 107 220)(93 234 108 219)(94 233 109 218)(95 232 110 217)(96 231 111 216)(97 230 112 215)(98 229 113 214)(99 228 114 213)(100 227 115 212)(101 226 116 211)(102 225 117 240)(103 224 118 239)(104 223 119 238)(105 222 120 237)(121 308 136 323)(122 307 137 322)(123 306 138 321)(124 305 139 320)(125 304 140 319)(126 303 141 318)(127 302 142 317)(128 301 143 316)(129 330 144 315)(130 329 145 314)(131 328 146 313)(132 327 147 312)(133 326 148 311)(134 325 149 310)(135 324 150 309)(151 335 166 350)(152 334 167 349)(153 333 168 348)(154 332 169 347)(155 331 170 346)(156 360 171 345)(157 359 172 344)(158 358 173 343)(159 357 174 342)(160 356 175 341)(161 355 176 340)(162 354 177 339)(163 353 178 338)(164 352 179 337)(165 351 180 336)(241 423 256 438)(242 422 257 437)(243 421 258 436)(244 450 259 435)(245 449 260 434)(246 448 261 433)(247 447 262 432)(248 446 263 431)(249 445 264 430)(250 444 265 429)(251 443 266 428)(252 442 267 427)(253 441 268 426)(254 440 269 425)(255 439 270 424)(271 476 286 461)(272 475 287 460)(273 474 288 459)(274 473 289 458)(275 472 290 457)(276 471 291 456)(277 470 292 455)(278 469 293 454)(279 468 294 453)(280 467 295 452)(281 466 296 451)(282 465 297 480)(283 464 298 479)(284 463 299 478)(285 462 300 477)(361 419 376 404)(362 418 377 403)(363 417 378 402)(364 416 379 401)(365 415 380 400)(366 414 381 399)(367 413 382 398)(368 412 383 397)(369 411 384 396)(370 410 385 395)(371 409 386 394)(372 408 387 393)(373 407 388 392)(374 406 389 391)(375 405 390 420)
(1 406 175 306 257 35 228 456)(2 417 176 317 258 46 229 467)(3 398 177 328 259 57 230 478)(4 409 178 309 260 38 231 459)(5 420 179 320 261 49 232 470)(6 401 180 301 262 60 233 451)(7 412 151 312 263 41 234 462)(8 393 152 323 264 52 235 473)(9 404 153 304 265 33 236 454)(10 415 154 315 266 44 237 465)(11 396 155 326 267 55 238 476)(12 407 156 307 268 36 239 457)(13 418 157 318 269 47 240 468)(14 399 158 329 270 58 211 479)(15 410 159 310 241 39 212 460)(16 391 160 321 242 50 213 471)(17 402 161 302 243 31 214 452)(18 413 162 313 244 42 215 463)(19 394 163 324 245 53 216 474)(20 405 164 305 246 34 217 455)(21 416 165 316 247 45 218 466)(22 397 166 327 248 56 219 477)(23 408 167 308 249 37 220 458)(24 419 168 319 250 48 221 469)(25 400 169 330 251 59 222 480)(26 411 170 311 252 40 223 461)(27 392 171 322 253 51 224 472)(28 403 172 303 254 32 225 453)(29 414 173 314 255 43 226 464)(30 395 174 325 256 54 227 475)(61 120 282 199 365 347 144 443)(62 101 283 210 366 358 145 424)(63 112 284 191 367 339 146 435)(64 93 285 202 368 350 147 446)(65 104 286 183 369 331 148 427)(66 115 287 194 370 342 149 438)(67 96 288 205 371 353 150 449)(68 107 289 186 372 334 121 430)(69 118 290 197 373 345 122 441)(70 99 291 208 374 356 123 422)(71 110 292 189 375 337 124 433)(72 91 293 200 376 348 125 444)(73 102 294 181 377 359 126 425)(74 113 295 192 378 340 127 436)(75 94 296 203 379 351 128 447)(76 105 297 184 380 332 129 428)(77 116 298 195 381 343 130 439)(78 97 299 206 382 354 131 450)(79 108 300 187 383 335 132 431)(80 119 271 198 384 346 133 442)(81 100 272 209 385 357 134 423)(82 111 273 190 386 338 135 434)(83 92 274 201 387 349 136 445)(84 103 275 182 388 360 137 426)(85 114 276 193 389 341 138 437)(86 95 277 204 390 352 139 448)(87 106 278 185 361 333 140 429)(88 117 279 196 362 344 141 440)(89 98 280 207 363 355 142 421)(90 109 281 188 364 336 143 432)
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), (1,193,16,208)(2,192,17,207)(3,191,18,206)(4,190,19,205)(5,189,20,204)(6,188,21,203)(7,187,22,202)(8,186,23,201)(9,185,24,200)(10,184,25,199)(11,183,26,198)(12,182,27,197)(13,181,28,196)(14,210,29,195)(15,209,30,194)(31,89,46,74)(32,88,47,73)(33,87,48,72)(34,86,49,71)(35,85,50,70)(36,84,51,69)(37,83,52,68)(38,82,53,67)(39,81,54,66)(40,80,55,65)(41,79,56,64)(42,78,57,63)(43,77,58,62)(44,76,59,61)(45,75,60,90)(91,236,106,221)(92,235,107,220)(93,234,108,219)(94,233,109,218)(95,232,110,217)(96,231,111,216)(97,230,112,215)(98,229,113,214)(99,228,114,213)(100,227,115,212)(101,226,116,211)(102,225,117,240)(103,224,118,239)(104,223,119,238)(105,222,120,237)(121,308,136,323)(122,307,137,322)(123,306,138,321)(124,305,139,320)(125,304,140,319)(126,303,141,318)(127,302,142,317)(128,301,143,316)(129,330,144,315)(130,329,145,314)(131,328,146,313)(132,327,147,312)(133,326,148,311)(134,325,149,310)(135,324,150,309)(151,335,166,350)(152,334,167,349)(153,333,168,348)(154,332,169,347)(155,331,170,346)(156,360,171,345)(157,359,172,344)(158,358,173,343)(159,357,174,342)(160,356,175,341)(161,355,176,340)(162,354,177,339)(163,353,178,338)(164,352,179,337)(165,351,180,336)(241,423,256,438)(242,422,257,437)(243,421,258,436)(244,450,259,435)(245,449,260,434)(246,448,261,433)(247,447,262,432)(248,446,263,431)(249,445,264,430)(250,444,265,429)(251,443,266,428)(252,442,267,427)(253,441,268,426)(254,440,269,425)(255,439,270,424)(271,476,286,461)(272,475,287,460)(273,474,288,459)(274,473,289,458)(275,472,290,457)(276,471,291,456)(277,470,292,455)(278,469,293,454)(279,468,294,453)(280,467,295,452)(281,466,296,451)(282,465,297,480)(283,464,298,479)(284,463,299,478)(285,462,300,477)(361,419,376,404)(362,418,377,403)(363,417,378,402)(364,416,379,401)(365,415,380,400)(366,414,381,399)(367,413,382,398)(368,412,383,397)(369,411,384,396)(370,410,385,395)(371,409,386,394)(372,408,387,393)(373,407,388,392)(374,406,389,391)(375,405,390,420), (1,406,175,306,257,35,228,456)(2,417,176,317,258,46,229,467)(3,398,177,328,259,57,230,478)(4,409,178,309,260,38,231,459)(5,420,179,320,261,49,232,470)(6,401,180,301,262,60,233,451)(7,412,151,312,263,41,234,462)(8,393,152,323,264,52,235,473)(9,404,153,304,265,33,236,454)(10,415,154,315,266,44,237,465)(11,396,155,326,267,55,238,476)(12,407,156,307,268,36,239,457)(13,418,157,318,269,47,240,468)(14,399,158,329,270,58,211,479)(15,410,159,310,241,39,212,460)(16,391,160,321,242,50,213,471)(17,402,161,302,243,31,214,452)(18,413,162,313,244,42,215,463)(19,394,163,324,245,53,216,474)(20,405,164,305,246,34,217,455)(21,416,165,316,247,45,218,466)(22,397,166,327,248,56,219,477)(23,408,167,308,249,37,220,458)(24,419,168,319,250,48,221,469)(25,400,169,330,251,59,222,480)(26,411,170,311,252,40,223,461)(27,392,171,322,253,51,224,472)(28,403,172,303,254,32,225,453)(29,414,173,314,255,43,226,464)(30,395,174,325,256,54,227,475)(61,120,282,199,365,347,144,443)(62,101,283,210,366,358,145,424)(63,112,284,191,367,339,146,435)(64,93,285,202,368,350,147,446)(65,104,286,183,369,331,148,427)(66,115,287,194,370,342,149,438)(67,96,288,205,371,353,150,449)(68,107,289,186,372,334,121,430)(69,118,290,197,373,345,122,441)(70,99,291,208,374,356,123,422)(71,110,292,189,375,337,124,433)(72,91,293,200,376,348,125,444)(73,102,294,181,377,359,126,425)(74,113,295,192,378,340,127,436)(75,94,296,203,379,351,128,447)(76,105,297,184,380,332,129,428)(77,116,298,195,381,343,130,439)(78,97,299,206,382,354,131,450)(79,108,300,187,383,335,132,431)(80,119,271,198,384,346,133,442)(81,100,272,209,385,357,134,423)(82,111,273,190,386,338,135,434)(83,92,274,201,387,349,136,445)(84,103,275,182,388,360,137,426)(85,114,276,193,389,341,138,437)(86,95,277,204,390,352,139,448)(87,106,278,185,361,333,140,429)(88,117,279,196,362,344,141,440)(89,98,280,207,363,355,142,421)(90,109,281,188,364,336,143,432)>;
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), (1,193,16,208)(2,192,17,207)(3,191,18,206)(4,190,19,205)(5,189,20,204)(6,188,21,203)(7,187,22,202)(8,186,23,201)(9,185,24,200)(10,184,25,199)(11,183,26,198)(12,182,27,197)(13,181,28,196)(14,210,29,195)(15,209,30,194)(31,89,46,74)(32,88,47,73)(33,87,48,72)(34,86,49,71)(35,85,50,70)(36,84,51,69)(37,83,52,68)(38,82,53,67)(39,81,54,66)(40,80,55,65)(41,79,56,64)(42,78,57,63)(43,77,58,62)(44,76,59,61)(45,75,60,90)(91,236,106,221)(92,235,107,220)(93,234,108,219)(94,233,109,218)(95,232,110,217)(96,231,111,216)(97,230,112,215)(98,229,113,214)(99,228,114,213)(100,227,115,212)(101,226,116,211)(102,225,117,240)(103,224,118,239)(104,223,119,238)(105,222,120,237)(121,308,136,323)(122,307,137,322)(123,306,138,321)(124,305,139,320)(125,304,140,319)(126,303,141,318)(127,302,142,317)(128,301,143,316)(129,330,144,315)(130,329,145,314)(131,328,146,313)(132,327,147,312)(133,326,148,311)(134,325,149,310)(135,324,150,309)(151,335,166,350)(152,334,167,349)(153,333,168,348)(154,332,169,347)(155,331,170,346)(156,360,171,345)(157,359,172,344)(158,358,173,343)(159,357,174,342)(160,356,175,341)(161,355,176,340)(162,354,177,339)(163,353,178,338)(164,352,179,337)(165,351,180,336)(241,423,256,438)(242,422,257,437)(243,421,258,436)(244,450,259,435)(245,449,260,434)(246,448,261,433)(247,447,262,432)(248,446,263,431)(249,445,264,430)(250,444,265,429)(251,443,266,428)(252,442,267,427)(253,441,268,426)(254,440,269,425)(255,439,270,424)(271,476,286,461)(272,475,287,460)(273,474,288,459)(274,473,289,458)(275,472,290,457)(276,471,291,456)(277,470,292,455)(278,469,293,454)(279,468,294,453)(280,467,295,452)(281,466,296,451)(282,465,297,480)(283,464,298,479)(284,463,299,478)(285,462,300,477)(361,419,376,404)(362,418,377,403)(363,417,378,402)(364,416,379,401)(365,415,380,400)(366,414,381,399)(367,413,382,398)(368,412,383,397)(369,411,384,396)(370,410,385,395)(371,409,386,394)(372,408,387,393)(373,407,388,392)(374,406,389,391)(375,405,390,420), (1,406,175,306,257,35,228,456)(2,417,176,317,258,46,229,467)(3,398,177,328,259,57,230,478)(4,409,178,309,260,38,231,459)(5,420,179,320,261,49,232,470)(6,401,180,301,262,60,233,451)(7,412,151,312,263,41,234,462)(8,393,152,323,264,52,235,473)(9,404,153,304,265,33,236,454)(10,415,154,315,266,44,237,465)(11,396,155,326,267,55,238,476)(12,407,156,307,268,36,239,457)(13,418,157,318,269,47,240,468)(14,399,158,329,270,58,211,479)(15,410,159,310,241,39,212,460)(16,391,160,321,242,50,213,471)(17,402,161,302,243,31,214,452)(18,413,162,313,244,42,215,463)(19,394,163,324,245,53,216,474)(20,405,164,305,246,34,217,455)(21,416,165,316,247,45,218,466)(22,397,166,327,248,56,219,477)(23,408,167,308,249,37,220,458)(24,419,168,319,250,48,221,469)(25,400,169,330,251,59,222,480)(26,411,170,311,252,40,223,461)(27,392,171,322,253,51,224,472)(28,403,172,303,254,32,225,453)(29,414,173,314,255,43,226,464)(30,395,174,325,256,54,227,475)(61,120,282,199,365,347,144,443)(62,101,283,210,366,358,145,424)(63,112,284,191,367,339,146,435)(64,93,285,202,368,350,147,446)(65,104,286,183,369,331,148,427)(66,115,287,194,370,342,149,438)(67,96,288,205,371,353,150,449)(68,107,289,186,372,334,121,430)(69,118,290,197,373,345,122,441)(70,99,291,208,374,356,123,422)(71,110,292,189,375,337,124,433)(72,91,293,200,376,348,125,444)(73,102,294,181,377,359,126,425)(74,113,295,192,378,340,127,436)(75,94,296,203,379,351,128,447)(76,105,297,184,380,332,129,428)(77,116,298,195,381,343,130,439)(78,97,299,206,382,354,131,450)(79,108,300,187,383,335,132,431)(80,119,271,198,384,346,133,442)(81,100,272,209,385,357,134,423)(82,111,273,190,386,338,135,434)(83,92,274,201,387,349,136,445)(84,103,275,182,388,360,137,426)(85,114,276,193,389,341,138,437)(86,95,277,204,390,352,139,448)(87,106,278,185,361,333,140,429)(88,117,279,196,362,344,141,440)(89,98,280,207,363,355,142,421)(90,109,281,188,364,336,143,432) );
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)], [(1,193,16,208),(2,192,17,207),(3,191,18,206),(4,190,19,205),(5,189,20,204),(6,188,21,203),(7,187,22,202),(8,186,23,201),(9,185,24,200),(10,184,25,199),(11,183,26,198),(12,182,27,197),(13,181,28,196),(14,210,29,195),(15,209,30,194),(31,89,46,74),(32,88,47,73),(33,87,48,72),(34,86,49,71),(35,85,50,70),(36,84,51,69),(37,83,52,68),(38,82,53,67),(39,81,54,66),(40,80,55,65),(41,79,56,64),(42,78,57,63),(43,77,58,62),(44,76,59,61),(45,75,60,90),(91,236,106,221),(92,235,107,220),(93,234,108,219),(94,233,109,218),(95,232,110,217),(96,231,111,216),(97,230,112,215),(98,229,113,214),(99,228,114,213),(100,227,115,212),(101,226,116,211),(102,225,117,240),(103,224,118,239),(104,223,119,238),(105,222,120,237),(121,308,136,323),(122,307,137,322),(123,306,138,321),(124,305,139,320),(125,304,140,319),(126,303,141,318),(127,302,142,317),(128,301,143,316),(129,330,144,315),(130,329,145,314),(131,328,146,313),(132,327,147,312),(133,326,148,311),(134,325,149,310),(135,324,150,309),(151,335,166,350),(152,334,167,349),(153,333,168,348),(154,332,169,347),(155,331,170,346),(156,360,171,345),(157,359,172,344),(158,358,173,343),(159,357,174,342),(160,356,175,341),(161,355,176,340),(162,354,177,339),(163,353,178,338),(164,352,179,337),(165,351,180,336),(241,423,256,438),(242,422,257,437),(243,421,258,436),(244,450,259,435),(245,449,260,434),(246,448,261,433),(247,447,262,432),(248,446,263,431),(249,445,264,430),(250,444,265,429),(251,443,266,428),(252,442,267,427),(253,441,268,426),(254,440,269,425),(255,439,270,424),(271,476,286,461),(272,475,287,460),(273,474,288,459),(274,473,289,458),(275,472,290,457),(276,471,291,456),(277,470,292,455),(278,469,293,454),(279,468,294,453),(280,467,295,452),(281,466,296,451),(282,465,297,480),(283,464,298,479),(284,463,299,478),(285,462,300,477),(361,419,376,404),(362,418,377,403),(363,417,378,402),(364,416,379,401),(365,415,380,400),(366,414,381,399),(367,413,382,398),(368,412,383,397),(369,411,384,396),(370,410,385,395),(371,409,386,394),(372,408,387,393),(373,407,388,392),(374,406,389,391),(375,405,390,420)], [(1,406,175,306,257,35,228,456),(2,417,176,317,258,46,229,467),(3,398,177,328,259,57,230,478),(4,409,178,309,260,38,231,459),(5,420,179,320,261,49,232,470),(6,401,180,301,262,60,233,451),(7,412,151,312,263,41,234,462),(8,393,152,323,264,52,235,473),(9,404,153,304,265,33,236,454),(10,415,154,315,266,44,237,465),(11,396,155,326,267,55,238,476),(12,407,156,307,268,36,239,457),(13,418,157,318,269,47,240,468),(14,399,158,329,270,58,211,479),(15,410,159,310,241,39,212,460),(16,391,160,321,242,50,213,471),(17,402,161,302,243,31,214,452),(18,413,162,313,244,42,215,463),(19,394,163,324,245,53,216,474),(20,405,164,305,246,34,217,455),(21,416,165,316,247,45,218,466),(22,397,166,327,248,56,219,477),(23,408,167,308,249,37,220,458),(24,419,168,319,250,48,221,469),(25,400,169,330,251,59,222,480),(26,411,170,311,252,40,223,461),(27,392,171,322,253,51,224,472),(28,403,172,303,254,32,225,453),(29,414,173,314,255,43,226,464),(30,395,174,325,256,54,227,475),(61,120,282,199,365,347,144,443),(62,101,283,210,366,358,145,424),(63,112,284,191,367,339,146,435),(64,93,285,202,368,350,147,446),(65,104,286,183,369,331,148,427),(66,115,287,194,370,342,149,438),(67,96,288,205,371,353,150,449),(68,107,289,186,372,334,121,430),(69,118,290,197,373,345,122,441),(70,99,291,208,374,356,123,422),(71,110,292,189,375,337,124,433),(72,91,293,200,376,348,125,444),(73,102,294,181,377,359,126,425),(74,113,295,192,378,340,127,436),(75,94,296,203,379,351,128,447),(76,105,297,184,380,332,129,428),(77,116,298,195,381,343,130,439),(78,97,299,206,382,354,131,450),(79,108,300,187,383,335,132,431),(80,119,271,198,384,346,133,442),(81,100,272,209,385,357,134,423),(82,111,273,190,386,338,135,434),(83,92,274,201,387,349,136,445),(84,103,275,182,388,360,137,426),(85,114,276,193,389,341,138,437),(86,95,277,204,390,352,139,448),(87,106,278,185,361,333,140,429),(88,117,279,196,362,344,141,440),(89,98,280,207,363,355,142,421),(90,109,281,188,364,336,143,432)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30F | 40A | ··· | 40P | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 15 | ··· | 15 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | - | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | S3 | D5 | Dic3 | D6 | Dic5 | D10 | C4×S3 | C4×S3 | C4×D5 | C4×D5 | S3×C8 | C8×D5 | S3×D5 | D5×Dic3 | S3×Dic5 | D30.C2 | D15⋊2C8 |
| kernel | Dic15⋊4C8 | C6×C5⋊2C8 | C10×C3⋊C8 | C4×Dic15 | C5×C3⋊C8 | C3×C5⋊2C8 | C2×Dic15 | Dic15 | C2×C5⋊2C8 | C2×C3⋊C8 | C5⋊2C8 | C2×C20 | C3⋊C8 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C10 | C6 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 16 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of Dic15⋊4C8 ►in GL4(𝔽241) generated by
| 52 | 240 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 182 | 15 |
| 0 | 177 | 0 | 0 |
| 177 | 0 | 0 | 0 |
| 0 | 0 | 232 | 66 |
| 0 | 0 | 28 | 9 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 169 | 46 |
| 0 | 0 | 224 | 72 |
G:=sub<GL(4,GF(241))| [52,1,0,0,240,0,0,0,0,0,225,182,0,0,0,15],[0,177,0,0,177,0,0,0,0,0,232,28,0,0,66,9],[8,0,0,0,0,8,0,0,0,0,169,224,0,0,46,72] >;
Dic15⋊4C8 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}\rtimes_4C_8 % in TeX
G:=Group("Dic15:4C8"); // GroupNames label
G:=SmallGroup(480,27);
// by ID
G=gap.SmallGroup(480,27);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,92,100,1356,18822]);
// 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^11,b*c=c*b>;
// generators/relations