metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic15.3Q8, C4⋊C4.5D15, C2.5(Q8×D15), C6.40(Q8×D5), (C2×C4).11D30, (C2×C20).37D6, C10.40(S3×Q8), C30.93(C2×Q8), C60⋊5C4.8C2, (C2×C12).36D10, C5⋊7(Dic3.Q8), C6.99(C4○D20), C15⋊16(C42.C2), C30.4Q8.3C2, C30.172(C4○D4), C10.99(C4○D12), C6.96(D4⋊2D5), (C2×C60).177C22, (C2×C30).286C23, (C4×Dic15).11C2, C3⋊7(Dic5.Q8), C2.11(D4⋊2D15), C10.96(D4⋊2S3), C2.13(D60⋊11C2), C22.47(C22×D15), (C2×Dic15).161C22, (C5×C4⋊C4).6S3, (C3×C4⋊C4).6D5, (C15×C4⋊C4).6C2, (C2×C6).282(C22×D5), (C2×C10).281(C22×S3), SmallGroup(480,854)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic15.3Q8
G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a15b, bd=db, dcd-1=a15c-1 >
Subgroups: 516 in 112 conjugacy classes, 49 normal (47 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C2×C4, C10, Dic3, C12, C2×C6, C15, C42, C4⋊C4, C4⋊C4, Dic5, C20, C2×C10, C2×Dic3, C2×C12, C30, C42.C2, C2×Dic5, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, Dic15, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, Dic3.Q8, C2×Dic15, C2×C60, Dic5.Q8, C4×Dic15, C30.4Q8, C60⋊5C4, C15×C4⋊C4, Dic15.3Q8
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, C22×S3, D15, C42.C2, C22×D5, C4○D12, D4⋊2S3, S3×Q8, D30, C4○D20, D4⋊2D5, Q8×D5, Dic3.Q8, C22×D15, Dic5.Q8, D60⋊11C2, D4⋊2D15, Q8×D15, Dic15.3Q8
►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)
(1 401 16 416)(2 400 17 415)(3 399 18 414)(4 398 19 413)(5 397 20 412)(6 396 21 411)(7 395 22 410)(8 394 23 409)(9 393 24 408)(10 392 25 407)(11 391 26 406)(12 420 27 405)(13 419 28 404)(14 418 29 403)(15 417 30 402)(31 305 46 320)(32 304 47 319)(33 303 48 318)(34 302 49 317)(35 301 50 316)(36 330 51 315)(37 329 52 314)(38 328 53 313)(39 327 54 312)(40 326 55 311)(41 325 56 310)(42 324 57 309)(43 323 58 308)(44 322 59 307)(45 321 60 306)(61 354 76 339)(62 353 77 338)(63 352 78 337)(64 351 79 336)(65 350 80 335)(66 349 81 334)(67 348 82 333)(68 347 83 332)(69 346 84 331)(70 345 85 360)(71 344 86 359)(72 343 87 358)(73 342 88 357)(74 341 89 356)(75 340 90 355)(91 207 106 192)(92 206 107 191)(93 205 108 190)(94 204 109 189)(95 203 110 188)(96 202 111 187)(97 201 112 186)(98 200 113 185)(99 199 114 184)(100 198 115 183)(101 197 116 182)(102 196 117 181)(103 195 118 210)(104 194 119 209)(105 193 120 208)(121 214 136 229)(122 213 137 228)(123 212 138 227)(124 211 139 226)(125 240 140 225)(126 239 141 224)(127 238 142 223)(128 237 143 222)(129 236 144 221)(130 235 145 220)(131 234 146 219)(132 233 147 218)(133 232 148 217)(134 231 149 216)(135 230 150 215)(151 368 166 383)(152 367 167 382)(153 366 168 381)(154 365 169 380)(155 364 170 379)(156 363 171 378)(157 362 172 377)(158 361 173 376)(159 390 174 375)(160 389 175 374)(161 388 176 373)(162 387 177 372)(163 386 178 371)(164 385 179 370)(165 384 180 369)(241 471 256 456)(242 470 257 455)(243 469 258 454)(244 468 259 453)(245 467 260 452)(246 466 261 451)(247 465 262 480)(248 464 263 479)(249 463 264 478)(250 462 265 477)(251 461 266 476)(252 460 267 475)(253 459 268 474)(254 458 269 473)(255 457 270 472)(271 428 286 443)(272 427 287 442)(273 426 288 441)(274 425 289 440)(275 424 290 439)(276 423 291 438)(277 422 292 437)(278 421 293 436)(279 450 294 435)(280 449 295 434)(281 448 296 433)(282 447 297 432)(283 446 298 431)(284 445 299 430)(285 444 300 429)
(1 167 106 459)(2 168 107 460)(3 169 108 461)(4 170 109 462)(5 171 110 463)(6 172 111 464)(7 173 112 465)(8 174 113 466)(9 175 114 467)(10 176 115 468)(11 177 116 469)(12 178 117 470)(13 179 118 471)(14 180 119 472)(15 151 120 473)(16 152 91 474)(17 153 92 475)(18 154 93 476)(19 155 94 477)(20 156 95 478)(21 157 96 479)(22 158 97 480)(23 159 98 451)(24 160 99 452)(25 161 100 453)(26 162 101 454)(27 163 102 455)(28 164 103 456)(29 165 104 457)(30 166 105 458)(31 426 135 353)(32 427 136 354)(33 428 137 355)(34 429 138 356)(35 430 139 357)(36 431 140 358)(37 432 141 359)(38 433 142 360)(39 434 143 331)(40 435 144 332)(41 436 145 333)(42 437 146 334)(43 438 147 335)(44 439 148 336)(45 440 149 337)(46 441 150 338)(47 442 121 339)(48 443 122 340)(49 444 123 341)(50 445 124 342)(51 446 125 343)(52 447 126 344)(53 448 127 345)(54 449 128 346)(55 450 129 347)(56 421 130 348)(57 422 131 349)(58 423 132 350)(59 424 133 351)(60 425 134 352)(61 304 272 229)(62 305 273 230)(63 306 274 231)(64 307 275 232)(65 308 276 233)(66 309 277 234)(67 310 278 235)(68 311 279 236)(69 312 280 237)(70 313 281 238)(71 314 282 239)(72 315 283 240)(73 316 284 211)(74 317 285 212)(75 318 286 213)(76 319 287 214)(77 320 288 215)(78 321 289 216)(79 322 290 217)(80 323 291 218)(81 324 292 219)(82 325 293 220)(83 326 294 221)(84 327 295 222)(85 328 296 223)(86 329 297 224)(87 330 298 225)(88 301 299 226)(89 302 300 227)(90 303 271 228)(181 242 420 386)(182 243 391 387)(183 244 392 388)(184 245 393 389)(185 246 394 390)(186 247 395 361)(187 248 396 362)(188 249 397 363)(189 250 398 364)(190 251 399 365)(191 252 400 366)(192 253 401 367)(193 254 402 368)(194 255 403 369)(195 256 404 370)(196 257 405 371)(197 258 406 372)(198 259 407 373)(199 260 408 374)(200 261 409 375)(201 262 410 376)(202 263 411 377)(203 264 412 378)(204 265 413 379)(205 266 414 380)(206 267 415 381)(207 268 416 382)(208 269 417 383)(209 270 418 384)(210 241 419 385)
(1 52 106 126)(2 51 107 125)(3 50 108 124)(4 49 109 123)(5 48 110 122)(6 47 111 121)(7 46 112 150)(8 45 113 149)(9 44 114 148)(10 43 115 147)(11 42 116 146)(12 41 117 145)(13 40 118 144)(14 39 119 143)(15 38 120 142)(16 37 91 141)(17 36 92 140)(18 35 93 139)(19 34 94 138)(20 33 95 137)(21 32 96 136)(22 31 97 135)(23 60 98 134)(24 59 99 133)(25 58 100 132)(26 57 101 131)(27 56 102 130)(28 55 103 129)(29 54 104 128)(30 53 105 127)(61 248 272 362)(62 247 273 361)(63 246 274 390)(64 245 275 389)(65 244 276 388)(66 243 277 387)(67 242 278 386)(68 241 279 385)(69 270 280 384)(70 269 281 383)(71 268 282 382)(72 267 283 381)(73 266 284 380)(74 265 285 379)(75 264 286 378)(76 263 287 377)(77 262 288 376)(78 261 289 375)(79 260 290 374)(80 259 291 373)(81 258 292 372)(82 257 293 371)(83 256 294 370)(84 255 295 369)(85 254 296 368)(86 253 297 367)(87 252 298 366)(88 251 299 365)(89 250 300 364)(90 249 271 363)(151 345 473 448)(152 344 474 447)(153 343 475 446)(154 342 476 445)(155 341 477 444)(156 340 478 443)(157 339 479 442)(158 338 480 441)(159 337 451 440)(160 336 452 439)(161 335 453 438)(162 334 454 437)(163 333 455 436)(164 332 456 435)(165 331 457 434)(166 360 458 433)(167 359 459 432)(168 358 460 431)(169 357 461 430)(170 356 462 429)(171 355 463 428)(172 354 464 427)(173 353 465 426)(174 352 466 425)(175 351 467 424)(176 350 468 423)(177 349 469 422)(178 348 470 421)(179 347 471 450)(180 346 472 449)(181 220 420 325)(182 219 391 324)(183 218 392 323)(184 217 393 322)(185 216 394 321)(186 215 395 320)(187 214 396 319)(188 213 397 318)(189 212 398 317)(190 211 399 316)(191 240 400 315)(192 239 401 314)(193 238 402 313)(194 237 403 312)(195 236 404 311)(196 235 405 310)(197 234 406 309)(198 233 407 308)(199 232 408 307)(200 231 409 306)(201 230 410 305)(202 229 411 304)(203 228 412 303)(204 227 413 302)(205 226 414 301)(206 225 415 330)(207 224 416 329)(208 223 417 328)(209 222 418 327)(210 221 419 326)
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), (1,401,16,416)(2,400,17,415)(3,399,18,414)(4,398,19,413)(5,397,20,412)(6,396,21,411)(7,395,22,410)(8,394,23,409)(9,393,24,408)(10,392,25,407)(11,391,26,406)(12,420,27,405)(13,419,28,404)(14,418,29,403)(15,417,30,402)(31,305,46,320)(32,304,47,319)(33,303,48,318)(34,302,49,317)(35,301,50,316)(36,330,51,315)(37,329,52,314)(38,328,53,313)(39,327,54,312)(40,326,55,311)(41,325,56,310)(42,324,57,309)(43,323,58,308)(44,322,59,307)(45,321,60,306)(61,354,76,339)(62,353,77,338)(63,352,78,337)(64,351,79,336)(65,350,80,335)(66,349,81,334)(67,348,82,333)(68,347,83,332)(69,346,84,331)(70,345,85,360)(71,344,86,359)(72,343,87,358)(73,342,88,357)(74,341,89,356)(75,340,90,355)(91,207,106,192)(92,206,107,191)(93,205,108,190)(94,204,109,189)(95,203,110,188)(96,202,111,187)(97,201,112,186)(98,200,113,185)(99,199,114,184)(100,198,115,183)(101,197,116,182)(102,196,117,181)(103,195,118,210)(104,194,119,209)(105,193,120,208)(121,214,136,229)(122,213,137,228)(123,212,138,227)(124,211,139,226)(125,240,140,225)(126,239,141,224)(127,238,142,223)(128,237,143,222)(129,236,144,221)(130,235,145,220)(131,234,146,219)(132,233,147,218)(133,232,148,217)(134,231,149,216)(135,230,150,215)(151,368,166,383)(152,367,167,382)(153,366,168,381)(154,365,169,380)(155,364,170,379)(156,363,171,378)(157,362,172,377)(158,361,173,376)(159,390,174,375)(160,389,175,374)(161,388,176,373)(162,387,177,372)(163,386,178,371)(164,385,179,370)(165,384,180,369)(241,471,256,456)(242,470,257,455)(243,469,258,454)(244,468,259,453)(245,467,260,452)(246,466,261,451)(247,465,262,480)(248,464,263,479)(249,463,264,478)(250,462,265,477)(251,461,266,476)(252,460,267,475)(253,459,268,474)(254,458,269,473)(255,457,270,472)(271,428,286,443)(272,427,287,442)(273,426,288,441)(274,425,289,440)(275,424,290,439)(276,423,291,438)(277,422,292,437)(278,421,293,436)(279,450,294,435)(280,449,295,434)(281,448,296,433)(282,447,297,432)(283,446,298,431)(284,445,299,430)(285,444,300,429), (1,167,106,459)(2,168,107,460)(3,169,108,461)(4,170,109,462)(5,171,110,463)(6,172,111,464)(7,173,112,465)(8,174,113,466)(9,175,114,467)(10,176,115,468)(11,177,116,469)(12,178,117,470)(13,179,118,471)(14,180,119,472)(15,151,120,473)(16,152,91,474)(17,153,92,475)(18,154,93,476)(19,155,94,477)(20,156,95,478)(21,157,96,479)(22,158,97,480)(23,159,98,451)(24,160,99,452)(25,161,100,453)(26,162,101,454)(27,163,102,455)(28,164,103,456)(29,165,104,457)(30,166,105,458)(31,426,135,353)(32,427,136,354)(33,428,137,355)(34,429,138,356)(35,430,139,357)(36,431,140,358)(37,432,141,359)(38,433,142,360)(39,434,143,331)(40,435,144,332)(41,436,145,333)(42,437,146,334)(43,438,147,335)(44,439,148,336)(45,440,149,337)(46,441,150,338)(47,442,121,339)(48,443,122,340)(49,444,123,341)(50,445,124,342)(51,446,125,343)(52,447,126,344)(53,448,127,345)(54,449,128,346)(55,450,129,347)(56,421,130,348)(57,422,131,349)(58,423,132,350)(59,424,133,351)(60,425,134,352)(61,304,272,229)(62,305,273,230)(63,306,274,231)(64,307,275,232)(65,308,276,233)(66,309,277,234)(67,310,278,235)(68,311,279,236)(69,312,280,237)(70,313,281,238)(71,314,282,239)(72,315,283,240)(73,316,284,211)(74,317,285,212)(75,318,286,213)(76,319,287,214)(77,320,288,215)(78,321,289,216)(79,322,290,217)(80,323,291,218)(81,324,292,219)(82,325,293,220)(83,326,294,221)(84,327,295,222)(85,328,296,223)(86,329,297,224)(87,330,298,225)(88,301,299,226)(89,302,300,227)(90,303,271,228)(181,242,420,386)(182,243,391,387)(183,244,392,388)(184,245,393,389)(185,246,394,390)(186,247,395,361)(187,248,396,362)(188,249,397,363)(189,250,398,364)(190,251,399,365)(191,252,400,366)(192,253,401,367)(193,254,402,368)(194,255,403,369)(195,256,404,370)(196,257,405,371)(197,258,406,372)(198,259,407,373)(199,260,408,374)(200,261,409,375)(201,262,410,376)(202,263,411,377)(203,264,412,378)(204,265,413,379)(205,266,414,380)(206,267,415,381)(207,268,416,382)(208,269,417,383)(209,270,418,384)(210,241,419,385), (1,52,106,126)(2,51,107,125)(3,50,108,124)(4,49,109,123)(5,48,110,122)(6,47,111,121)(7,46,112,150)(8,45,113,149)(9,44,114,148)(10,43,115,147)(11,42,116,146)(12,41,117,145)(13,40,118,144)(14,39,119,143)(15,38,120,142)(16,37,91,141)(17,36,92,140)(18,35,93,139)(19,34,94,138)(20,33,95,137)(21,32,96,136)(22,31,97,135)(23,60,98,134)(24,59,99,133)(25,58,100,132)(26,57,101,131)(27,56,102,130)(28,55,103,129)(29,54,104,128)(30,53,105,127)(61,248,272,362)(62,247,273,361)(63,246,274,390)(64,245,275,389)(65,244,276,388)(66,243,277,387)(67,242,278,386)(68,241,279,385)(69,270,280,384)(70,269,281,383)(71,268,282,382)(72,267,283,381)(73,266,284,380)(74,265,285,379)(75,264,286,378)(76,263,287,377)(77,262,288,376)(78,261,289,375)(79,260,290,374)(80,259,291,373)(81,258,292,372)(82,257,293,371)(83,256,294,370)(84,255,295,369)(85,254,296,368)(86,253,297,367)(87,252,298,366)(88,251,299,365)(89,250,300,364)(90,249,271,363)(151,345,473,448)(152,344,474,447)(153,343,475,446)(154,342,476,445)(155,341,477,444)(156,340,478,443)(157,339,479,442)(158,338,480,441)(159,337,451,440)(160,336,452,439)(161,335,453,438)(162,334,454,437)(163,333,455,436)(164,332,456,435)(165,331,457,434)(166,360,458,433)(167,359,459,432)(168,358,460,431)(169,357,461,430)(170,356,462,429)(171,355,463,428)(172,354,464,427)(173,353,465,426)(174,352,466,425)(175,351,467,424)(176,350,468,423)(177,349,469,422)(178,348,470,421)(179,347,471,450)(180,346,472,449)(181,220,420,325)(182,219,391,324)(183,218,392,323)(184,217,393,322)(185,216,394,321)(186,215,395,320)(187,214,396,319)(188,213,397,318)(189,212,398,317)(190,211,399,316)(191,240,400,315)(192,239,401,314)(193,238,402,313)(194,237,403,312)(195,236,404,311)(196,235,405,310)(197,234,406,309)(198,233,407,308)(199,232,408,307)(200,231,409,306)(201,230,410,305)(202,229,411,304)(203,228,412,303)(204,227,413,302)(205,226,414,301)(206,225,415,330)(207,224,416,329)(208,223,417,328)(209,222,418,327)(210,221,419,326)>;
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), (1,401,16,416)(2,400,17,415)(3,399,18,414)(4,398,19,413)(5,397,20,412)(6,396,21,411)(7,395,22,410)(8,394,23,409)(9,393,24,408)(10,392,25,407)(11,391,26,406)(12,420,27,405)(13,419,28,404)(14,418,29,403)(15,417,30,402)(31,305,46,320)(32,304,47,319)(33,303,48,318)(34,302,49,317)(35,301,50,316)(36,330,51,315)(37,329,52,314)(38,328,53,313)(39,327,54,312)(40,326,55,311)(41,325,56,310)(42,324,57,309)(43,323,58,308)(44,322,59,307)(45,321,60,306)(61,354,76,339)(62,353,77,338)(63,352,78,337)(64,351,79,336)(65,350,80,335)(66,349,81,334)(67,348,82,333)(68,347,83,332)(69,346,84,331)(70,345,85,360)(71,344,86,359)(72,343,87,358)(73,342,88,357)(74,341,89,356)(75,340,90,355)(91,207,106,192)(92,206,107,191)(93,205,108,190)(94,204,109,189)(95,203,110,188)(96,202,111,187)(97,201,112,186)(98,200,113,185)(99,199,114,184)(100,198,115,183)(101,197,116,182)(102,196,117,181)(103,195,118,210)(104,194,119,209)(105,193,120,208)(121,214,136,229)(122,213,137,228)(123,212,138,227)(124,211,139,226)(125,240,140,225)(126,239,141,224)(127,238,142,223)(128,237,143,222)(129,236,144,221)(130,235,145,220)(131,234,146,219)(132,233,147,218)(133,232,148,217)(134,231,149,216)(135,230,150,215)(151,368,166,383)(152,367,167,382)(153,366,168,381)(154,365,169,380)(155,364,170,379)(156,363,171,378)(157,362,172,377)(158,361,173,376)(159,390,174,375)(160,389,175,374)(161,388,176,373)(162,387,177,372)(163,386,178,371)(164,385,179,370)(165,384,180,369)(241,471,256,456)(242,470,257,455)(243,469,258,454)(244,468,259,453)(245,467,260,452)(246,466,261,451)(247,465,262,480)(248,464,263,479)(249,463,264,478)(250,462,265,477)(251,461,266,476)(252,460,267,475)(253,459,268,474)(254,458,269,473)(255,457,270,472)(271,428,286,443)(272,427,287,442)(273,426,288,441)(274,425,289,440)(275,424,290,439)(276,423,291,438)(277,422,292,437)(278,421,293,436)(279,450,294,435)(280,449,295,434)(281,448,296,433)(282,447,297,432)(283,446,298,431)(284,445,299,430)(285,444,300,429), (1,167,106,459)(2,168,107,460)(3,169,108,461)(4,170,109,462)(5,171,110,463)(6,172,111,464)(7,173,112,465)(8,174,113,466)(9,175,114,467)(10,176,115,468)(11,177,116,469)(12,178,117,470)(13,179,118,471)(14,180,119,472)(15,151,120,473)(16,152,91,474)(17,153,92,475)(18,154,93,476)(19,155,94,477)(20,156,95,478)(21,157,96,479)(22,158,97,480)(23,159,98,451)(24,160,99,452)(25,161,100,453)(26,162,101,454)(27,163,102,455)(28,164,103,456)(29,165,104,457)(30,166,105,458)(31,426,135,353)(32,427,136,354)(33,428,137,355)(34,429,138,356)(35,430,139,357)(36,431,140,358)(37,432,141,359)(38,433,142,360)(39,434,143,331)(40,435,144,332)(41,436,145,333)(42,437,146,334)(43,438,147,335)(44,439,148,336)(45,440,149,337)(46,441,150,338)(47,442,121,339)(48,443,122,340)(49,444,123,341)(50,445,124,342)(51,446,125,343)(52,447,126,344)(53,448,127,345)(54,449,128,346)(55,450,129,347)(56,421,130,348)(57,422,131,349)(58,423,132,350)(59,424,133,351)(60,425,134,352)(61,304,272,229)(62,305,273,230)(63,306,274,231)(64,307,275,232)(65,308,276,233)(66,309,277,234)(67,310,278,235)(68,311,279,236)(69,312,280,237)(70,313,281,238)(71,314,282,239)(72,315,283,240)(73,316,284,211)(74,317,285,212)(75,318,286,213)(76,319,287,214)(77,320,288,215)(78,321,289,216)(79,322,290,217)(80,323,291,218)(81,324,292,219)(82,325,293,220)(83,326,294,221)(84,327,295,222)(85,328,296,223)(86,329,297,224)(87,330,298,225)(88,301,299,226)(89,302,300,227)(90,303,271,228)(181,242,420,386)(182,243,391,387)(183,244,392,388)(184,245,393,389)(185,246,394,390)(186,247,395,361)(187,248,396,362)(188,249,397,363)(189,250,398,364)(190,251,399,365)(191,252,400,366)(192,253,401,367)(193,254,402,368)(194,255,403,369)(195,256,404,370)(196,257,405,371)(197,258,406,372)(198,259,407,373)(199,260,408,374)(200,261,409,375)(201,262,410,376)(202,263,411,377)(203,264,412,378)(204,265,413,379)(205,266,414,380)(206,267,415,381)(207,268,416,382)(208,269,417,383)(209,270,418,384)(210,241,419,385), (1,52,106,126)(2,51,107,125)(3,50,108,124)(4,49,109,123)(5,48,110,122)(6,47,111,121)(7,46,112,150)(8,45,113,149)(9,44,114,148)(10,43,115,147)(11,42,116,146)(12,41,117,145)(13,40,118,144)(14,39,119,143)(15,38,120,142)(16,37,91,141)(17,36,92,140)(18,35,93,139)(19,34,94,138)(20,33,95,137)(21,32,96,136)(22,31,97,135)(23,60,98,134)(24,59,99,133)(25,58,100,132)(26,57,101,131)(27,56,102,130)(28,55,103,129)(29,54,104,128)(30,53,105,127)(61,248,272,362)(62,247,273,361)(63,246,274,390)(64,245,275,389)(65,244,276,388)(66,243,277,387)(67,242,278,386)(68,241,279,385)(69,270,280,384)(70,269,281,383)(71,268,282,382)(72,267,283,381)(73,266,284,380)(74,265,285,379)(75,264,286,378)(76,263,287,377)(77,262,288,376)(78,261,289,375)(79,260,290,374)(80,259,291,373)(81,258,292,372)(82,257,293,371)(83,256,294,370)(84,255,295,369)(85,254,296,368)(86,253,297,367)(87,252,298,366)(88,251,299,365)(89,250,300,364)(90,249,271,363)(151,345,473,448)(152,344,474,447)(153,343,475,446)(154,342,476,445)(155,341,477,444)(156,340,478,443)(157,339,479,442)(158,338,480,441)(159,337,451,440)(160,336,452,439)(161,335,453,438)(162,334,454,437)(163,333,455,436)(164,332,456,435)(165,331,457,434)(166,360,458,433)(167,359,459,432)(168,358,460,431)(169,357,461,430)(170,356,462,429)(171,355,463,428)(172,354,464,427)(173,353,465,426)(174,352,466,425)(175,351,467,424)(176,350,468,423)(177,349,469,422)(178,348,470,421)(179,347,471,450)(180,346,472,449)(181,220,420,325)(182,219,391,324)(183,218,392,323)(184,217,393,322)(185,216,394,321)(186,215,395,320)(187,214,396,319)(188,213,397,318)(189,212,398,317)(190,211,399,316)(191,240,400,315)(192,239,401,314)(193,238,402,313)(194,237,403,312)(195,236,404,311)(196,235,405,310)(197,234,406,309)(198,233,407,308)(199,232,408,307)(200,231,409,306)(201,230,410,305)(202,229,411,304)(203,228,412,303)(204,227,413,302)(205,226,414,301)(206,225,415,330)(207,224,416,329)(208,223,417,328)(209,222,418,327)(210,221,419,326) );
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)], [(1,401,16,416),(2,400,17,415),(3,399,18,414),(4,398,19,413),(5,397,20,412),(6,396,21,411),(7,395,22,410),(8,394,23,409),(9,393,24,408),(10,392,25,407),(11,391,26,406),(12,420,27,405),(13,419,28,404),(14,418,29,403),(15,417,30,402),(31,305,46,320),(32,304,47,319),(33,303,48,318),(34,302,49,317),(35,301,50,316),(36,330,51,315),(37,329,52,314),(38,328,53,313),(39,327,54,312),(40,326,55,311),(41,325,56,310),(42,324,57,309),(43,323,58,308),(44,322,59,307),(45,321,60,306),(61,354,76,339),(62,353,77,338),(63,352,78,337),(64,351,79,336),(65,350,80,335),(66,349,81,334),(67,348,82,333),(68,347,83,332),(69,346,84,331),(70,345,85,360),(71,344,86,359),(72,343,87,358),(73,342,88,357),(74,341,89,356),(75,340,90,355),(91,207,106,192),(92,206,107,191),(93,205,108,190),(94,204,109,189),(95,203,110,188),(96,202,111,187),(97,201,112,186),(98,200,113,185),(99,199,114,184),(100,198,115,183),(101,197,116,182),(102,196,117,181),(103,195,118,210),(104,194,119,209),(105,193,120,208),(121,214,136,229),(122,213,137,228),(123,212,138,227),(124,211,139,226),(125,240,140,225),(126,239,141,224),(127,238,142,223),(128,237,143,222),(129,236,144,221),(130,235,145,220),(131,234,146,219),(132,233,147,218),(133,232,148,217),(134,231,149,216),(135,230,150,215),(151,368,166,383),(152,367,167,382),(153,366,168,381),(154,365,169,380),(155,364,170,379),(156,363,171,378),(157,362,172,377),(158,361,173,376),(159,390,174,375),(160,389,175,374),(161,388,176,373),(162,387,177,372),(163,386,178,371),(164,385,179,370),(165,384,180,369),(241,471,256,456),(242,470,257,455),(243,469,258,454),(244,468,259,453),(245,467,260,452),(246,466,261,451),(247,465,262,480),(248,464,263,479),(249,463,264,478),(250,462,265,477),(251,461,266,476),(252,460,267,475),(253,459,268,474),(254,458,269,473),(255,457,270,472),(271,428,286,443),(272,427,287,442),(273,426,288,441),(274,425,289,440),(275,424,290,439),(276,423,291,438),(277,422,292,437),(278,421,293,436),(279,450,294,435),(280,449,295,434),(281,448,296,433),(282,447,297,432),(283,446,298,431),(284,445,299,430),(285,444,300,429)], [(1,167,106,459),(2,168,107,460),(3,169,108,461),(4,170,109,462),(5,171,110,463),(6,172,111,464),(7,173,112,465),(8,174,113,466),(9,175,114,467),(10,176,115,468),(11,177,116,469),(12,178,117,470),(13,179,118,471),(14,180,119,472),(15,151,120,473),(16,152,91,474),(17,153,92,475),(18,154,93,476),(19,155,94,477),(20,156,95,478),(21,157,96,479),(22,158,97,480),(23,159,98,451),(24,160,99,452),(25,161,100,453),(26,162,101,454),(27,163,102,455),(28,164,103,456),(29,165,104,457),(30,166,105,458),(31,426,135,353),(32,427,136,354),(33,428,137,355),(34,429,138,356),(35,430,139,357),(36,431,140,358),(37,432,141,359),(38,433,142,360),(39,434,143,331),(40,435,144,332),(41,436,145,333),(42,437,146,334),(43,438,147,335),(44,439,148,336),(45,440,149,337),(46,441,150,338),(47,442,121,339),(48,443,122,340),(49,444,123,341),(50,445,124,342),(51,446,125,343),(52,447,126,344),(53,448,127,345),(54,449,128,346),(55,450,129,347),(56,421,130,348),(57,422,131,349),(58,423,132,350),(59,424,133,351),(60,425,134,352),(61,304,272,229),(62,305,273,230),(63,306,274,231),(64,307,275,232),(65,308,276,233),(66,309,277,234),(67,310,278,235),(68,311,279,236),(69,312,280,237),(70,313,281,238),(71,314,282,239),(72,315,283,240),(73,316,284,211),(74,317,285,212),(75,318,286,213),(76,319,287,214),(77,320,288,215),(78,321,289,216),(79,322,290,217),(80,323,291,218),(81,324,292,219),(82,325,293,220),(83,326,294,221),(84,327,295,222),(85,328,296,223),(86,329,297,224),(87,330,298,225),(88,301,299,226),(89,302,300,227),(90,303,271,228),(181,242,420,386),(182,243,391,387),(183,244,392,388),(184,245,393,389),(185,246,394,390),(186,247,395,361),(187,248,396,362),(188,249,397,363),(189,250,398,364),(190,251,399,365),(191,252,400,366),(192,253,401,367),(193,254,402,368),(194,255,403,369),(195,256,404,370),(196,257,405,371),(197,258,406,372),(198,259,407,373),(199,260,408,374),(200,261,409,375),(201,262,410,376),(202,263,411,377),(203,264,412,378),(204,265,413,379),(205,266,414,380),(206,267,415,381),(207,268,416,382),(208,269,417,383),(209,270,418,384),(210,241,419,385)], [(1,52,106,126),(2,51,107,125),(3,50,108,124),(4,49,109,123),(5,48,110,122),(6,47,111,121),(7,46,112,150),(8,45,113,149),(9,44,114,148),(10,43,115,147),(11,42,116,146),(12,41,117,145),(13,40,118,144),(14,39,119,143),(15,38,120,142),(16,37,91,141),(17,36,92,140),(18,35,93,139),(19,34,94,138),(20,33,95,137),(21,32,96,136),(22,31,97,135),(23,60,98,134),(24,59,99,133),(25,58,100,132),(26,57,101,131),(27,56,102,130),(28,55,103,129),(29,54,104,128),(30,53,105,127),(61,248,272,362),(62,247,273,361),(63,246,274,390),(64,245,275,389),(65,244,276,388),(66,243,277,387),(67,242,278,386),(68,241,279,385),(69,270,280,384),(70,269,281,383),(71,268,282,382),(72,267,283,381),(73,266,284,380),(74,265,285,379),(75,264,286,378),(76,263,287,377),(77,262,288,376),(78,261,289,375),(79,260,290,374),(80,259,291,373),(81,258,292,372),(82,257,293,371),(83,256,294,370),(84,255,295,369),(85,254,296,368),(86,253,297,367),(87,252,298,366),(88,251,299,365),(89,250,300,364),(90,249,271,363),(151,345,473,448),(152,344,474,447),(153,343,475,446),(154,342,476,445),(155,341,477,444),(156,340,478,443),(157,339,479,442),(158,338,480,441),(159,337,451,440),(160,336,452,439),(161,335,453,438),(162,334,454,437),(163,333,455,436),(164,332,456,435),(165,331,457,434),(166,360,458,433),(167,359,459,432),(168,358,460,431),(169,357,461,430),(170,356,462,429),(171,355,463,428),(172,354,464,427),(173,353,465,426),(174,352,466,425),(175,351,467,424),(176,350,468,423),(177,349,469,422),(178,348,470,421),(179,347,471,450),(180,346,472,449),(181,220,420,325),(182,219,391,324),(183,218,392,323),(184,217,393,322),(185,216,394,321),(186,215,395,320),(187,214,396,319),(188,213,397,318),(189,212,398,317),(190,211,399,316),(191,240,400,315),(192,239,401,314),(193,238,402,313),(194,237,403,312),(195,236,404,311),(196,235,405,310),(197,234,406,309),(198,233,407,308),(199,232,408,307),(200,231,409,306),(201,230,410,305),(202,229,411,304),(203,228,412,303),(204,227,413,302),(205,226,414,301),(206,225,415,330),(207,224,416,329),(208,223,417,328),(209,222,418,327),(210,221,419,326)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 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 | 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 | 4 | 4 | 30 | 30 | 30 | 30 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 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 | S3 | Q8 | D5 | D6 | C4○D4 | D10 | D15 | C4○D12 | D30 | C4○D20 | D60⋊11C2 | D4⋊2S3 | S3×Q8 | D4⋊2D5 | Q8×D5 | D4⋊2D15 | Q8×D15 |
| kernel | Dic15.3Q8 | C4×Dic15 | C30.4Q8 | C60⋊5C4 | C15×C4⋊C4 | C5×C4⋊C4 | Dic15 | C3×C4⋊C4 | C2×C20 | C30 | C2×C12 | C4⋊C4 | C10 | C2×C4 | C6 | C2 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 6 | 4 | 4 | 12 | 8 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of Dic15.3Q8 ►in GL6(𝔽61)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 59 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 42 |
| 0 | 0 | 0 | 0 | 30 | 52 |
G:=sub<GL(6,GF(61))| [16,0,0,0,0,0,0,42,0,0,0,0,0,0,27,0,0,0,0,0,0,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,50,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,59,60],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,30,0,0,0,0,42,52] >;
Dic15.3Q8 in GAP, Magma, Sage, TeX
{\rm Dic}_{15}._3Q_8 % in TeX
G:=Group("Dic15.3Q8"); // GroupNames label
G:=SmallGroup(480,854);
// by ID
G=gap.SmallGroup(480,854);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,254,219,100,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=c^4=1,b^2=a^15,d^2=c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=a^15*c^-1>;
// generators/relations