metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.7Dic6, C6.30(Q8×D5), C30.27(C2×Q8), (C2×C20).186D6, Dic3⋊C4.1D5, (C3×Dic5).7Q8, C2.14(D5×Dic6), C5⋊1(C12.6Q8), C15⋊7(C42.C2), C6.29(C4○D20), (C2×C12).260D10, (C2×C30).68C23, (C4×Dic5).11S3, C10.12(C2×Dic6), C10.32(C4○D12), C30.107(C4○D4), C6.40(D4⋊2D5), (C2×C60).383C22, (C2×Dic3).21D10, (C2×Dic5).167D6, (C12×Dic5).23C2, C3⋊1(Dic5.Q8), C30.4Q8.13C2, Dic15⋊5C4.10C2, C6.Dic10.10C2, C30.Q8.11C2, C2.14(Dic3.D10), C2.21(D6.D10), (C2×Dic15).62C22, (C6×Dic5).189C22, (C10×Dic3).40C22, (C2×C4).123(S3×D5), C22.154(C2×S3×D5), (C2×C6).80(C22×D5), (C5×Dic3⋊C4).13C2, (C2×C10).80(C22×S3), SmallGroup(480,454)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.7Dic6
G = < a,b,c,d | a10=c12=1, b2=a5, d2=c6, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a5b, dcd-1=a5c-1 >
Subgroups: 460 in 112 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C2×C4, C10, Dic3, C12, C2×C6, C15, C42, C4⋊C4, Dic5, Dic5, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4×C12, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C12.6Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, Dic5.Q8, C30.Q8, Dic15⋊5C4, C6.Dic10, C12×Dic5, C5×Dic3⋊C4, C30.4Q8, Dic5.7Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, Dic6, C22×S3, C42.C2, C22×D5, C2×Dic6, C4○D12, S3×D5, C4○D20, D4⋊2D5, Q8×D5, C12.6Q8, C2×S3×D5, Dic5.Q8, D5×Dic6, D6.D10, Dic3.D10, Dic5.7Dic6
►Regular action on 480 points
Generators in S480
(1 335 118 366 19 242 210 293 304 358)(2 359 305 294 211 243 20 367 119 336)(3 325 120 368 21 244 212 295 306 360)(4 349 307 296 213 245 22 369 109 326)(5 327 110 370 23 246 214 297 308 350)(6 351 309 298 215 247 24 371 111 328)(7 329 112 372 13 248 216 299 310 352)(8 353 311 300 205 249 14 361 113 330)(9 331 114 362 15 250 206 289 312 354)(10 355 301 290 207 251 16 363 115 332)(11 333 116 364 17 252 208 291 302 356)(12 357 303 292 209 241 18 365 117 334)(25 188 47 162 73 376 342 282 65 396)(26 385 66 283 343 377 74 163 48 189)(27 190 37 164 75 378 344 284 67 386)(28 387 68 285 345 379 76 165 38 191)(29 192 39 166 77 380 346 286 69 388)(30 389 70 287 347 381 78 167 40 181)(31 182 41 168 79 382 348 288 71 390)(32 391 72 277 337 383 80 157 42 183)(33 184 43 158 81 384 338 278 61 392)(34 393 62 279 339 373 82 159 44 185)(35 186 45 160 83 374 340 280 63 394)(36 395 64 281 341 375 84 161 46 187)(49 456 410 228 406 93 424 230 315 257)(50 258 316 231 425 94 407 217 411 445)(51 446 412 218 408 95 426 232 317 259)(52 260 318 233 427 96 397 219 413 447)(53 448 414 220 398 85 428 234 319 261)(54 262 320 235 429 86 399 221 415 449)(55 450 416 222 400 87 430 236 321 263)(56 264 322 237 431 88 401 223 417 451)(57 452 418 224 402 89 432 238 323 253)(58 254 324 239 421 90 403 225 419 453)(59 454 420 226 404 91 422 240 313 255)(60 256 314 229 423 92 405 227 409 455)(97 204 126 466 266 480 439 149 137 170)(98 171 138 150 440 469 267 467 127 193)(99 194 128 468 268 470 441 151 139 172)(100 173 140 152 442 471 269 457 129 195)(101 196 130 458 270 472 443 153 141 174)(102 175 142 154 444 473 271 459 131 197)(103 198 132 460 272 474 433 155 143 176)(104 177 144 156 434 475 273 461 121 199)(105 200 122 462 274 476 435 145 133 178)(106 179 134 146 436 477 275 463 123 201)(107 202 124 464 276 478 437 147 135 180)(108 169 136 148 438 479 265 465 125 203)
(1 427 242 447)(2 428 243 448)(3 429 244 449)(4 430 245 450)(5 431 246 451)(6 432 247 452)(7 421 248 453)(8 422 249 454)(9 423 250 455)(10 424 251 456)(11 425 252 445)(12 426 241 446)(13 58 352 90)(14 59 353 91)(15 60 354 92)(16 49 355 93)(17 50 356 94)(18 51 357 95)(19 52 358 96)(20 53 359 85)(21 54 360 86)(22 55 349 87)(23 56 350 88)(24 57 351 89)(25 476 376 105)(26 477 377 106)(27 478 378 107)(28 479 379 108)(29 480 380 97)(30 469 381 98)(31 470 382 99)(32 471 383 100)(33 472 384 101)(34 473 373 102)(35 474 374 103)(36 475 375 104)(37 464 284 135)(38 465 285 136)(39 466 286 137)(40 467 287 138)(41 468 288 139)(42 457 277 140)(43 458 278 141)(44 459 279 142)(45 460 280 143)(46 461 281 144)(47 462 282 133)(48 463 283 134)(61 153 158 130)(62 154 159 131)(63 155 160 132)(64 156 161 121)(65 145 162 122)(66 146 163 123)(67 147 164 124)(68 148 165 125)(69 149 166 126)(70 150 167 127)(71 151 168 128)(72 152 157 129)(73 200 396 435)(74 201 385 436)(75 202 386 437)(76 203 387 438)(77 204 388 439)(78 193 389 440)(79 194 390 441)(80 195 391 442)(81 196 392 443)(82 197 393 444)(83 198 394 433)(84 199 395 434)(109 321 296 222)(110 322 297 223)(111 323 298 224)(112 324 299 225)(113 313 300 226)(114 314 289 227)(115 315 290 228)(116 316 291 217)(117 317 292 218)(118 318 293 219)(119 319 294 220)(120 320 295 221)(169 191 265 345)(170 192 266 346)(171 181 267 347)(172 182 268 348)(173 183 269 337)(174 184 270 338)(175 185 271 339)(176 186 272 340)(177 187 273 341)(178 188 274 342)(179 189 275 343)(180 190 276 344)(205 420 330 240)(206 409 331 229)(207 410 332 230)(208 411 333 231)(209 412 334 232)(210 413 335 233)(211 414 336 234)(212 415 325 235)(213 416 326 236)(214 417 327 237)(215 418 328 238)(216 419 329 239)(253 309 402 371)(254 310 403 372)(255 311 404 361)(256 312 405 362)(257 301 406 363)(258 302 407 364)(259 303 408 365)(260 304 397 366)(261 305 398 367)(262 306 399 368)(263 307 400 369)(264 308 401 370)
(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)
(1 388 7 394)(2 76 8 82)(3 386 9 392)(4 74 10 80)(5 396 11 390)(6 84 12 78)(13 160 19 166)(14 62 20 68)(15 158 21 164)(16 72 22 66)(17 168 23 162)(18 70 24 64)(25 333 31 327)(26 207 32 213)(27 331 33 325)(28 205 34 211)(29 329 35 335)(30 215 36 209)(37 362 43 368)(38 311 44 305)(39 372 45 366)(40 309 46 303)(41 370 47 364)(42 307 48 301)(49 129 55 123)(50 151 56 145)(51 127 57 121)(52 149 58 155)(53 125 59 131)(54 147 60 153)(61 360 67 354)(63 358 69 352)(65 356 71 350)(73 252 79 246)(75 250 81 244)(77 248 83 242)(85 148 91 154)(86 124 92 130)(87 146 93 152)(88 122 94 128)(89 156 95 150)(90 132 96 126)(97 239 103 233)(98 418 104 412)(99 237 105 231)(100 416 106 410)(101 235 107 229)(102 414 108 420)(109 343 115 337)(110 188 116 182)(111 341 117 347)(112 186 118 192)(113 339 119 345)(114 184 120 190)(133 258 139 264)(134 406 140 400)(135 256 141 262)(136 404 142 398)(137 254 143 260)(138 402 144 408)(157 349 163 355)(159 359 165 353)(161 357 167 351)(169 226 175 220)(170 324 176 318)(171 224 177 218)(172 322 178 316)(173 222 179 228)(174 320 180 314)(181 298 187 292)(183 296 189 290)(185 294 191 300)(193 452 199 446)(194 431 200 425)(195 450 201 456)(196 429 202 423)(197 448 203 454)(198 427 204 421)(206 384 212 378)(208 382 214 376)(210 380 216 374)(217 268 223 274)(219 266 225 272)(221 276 227 270)(230 471 236 477)(232 469 238 475)(234 479 240 473)(241 389 247 395)(243 387 249 393)(245 385 251 391)(253 461 259 467)(255 459 261 465)(257 457 263 463)(265 313 271 319)(267 323 273 317)(269 321 275 315)(277 369 283 363)(278 306 284 312)(279 367 285 361)(280 304 286 310)(281 365 287 371)(282 302 288 308)(289 338 295 344)(291 348 297 342)(293 346 299 340)(326 377 332 383)(328 375 334 381)(330 373 336 379)(397 466 403 460)(399 464 405 458)(401 462 407 468)(409 472 415 478)(411 470 417 476)(413 480 419 474)(422 444 428 438)(424 442 430 436)(426 440 432 434)(433 447 439 453)(435 445 441 451)(437 455 443 449)
G:=sub<Sym(480)| (1,335,118,366,19,242,210,293,304,358)(2,359,305,294,211,243,20,367,119,336)(3,325,120,368,21,244,212,295,306,360)(4,349,307,296,213,245,22,369,109,326)(5,327,110,370,23,246,214,297,308,350)(6,351,309,298,215,247,24,371,111,328)(7,329,112,372,13,248,216,299,310,352)(8,353,311,300,205,249,14,361,113,330)(9,331,114,362,15,250,206,289,312,354)(10,355,301,290,207,251,16,363,115,332)(11,333,116,364,17,252,208,291,302,356)(12,357,303,292,209,241,18,365,117,334)(25,188,47,162,73,376,342,282,65,396)(26,385,66,283,343,377,74,163,48,189)(27,190,37,164,75,378,344,284,67,386)(28,387,68,285,345,379,76,165,38,191)(29,192,39,166,77,380,346,286,69,388)(30,389,70,287,347,381,78,167,40,181)(31,182,41,168,79,382,348,288,71,390)(32,391,72,277,337,383,80,157,42,183)(33,184,43,158,81,384,338,278,61,392)(34,393,62,279,339,373,82,159,44,185)(35,186,45,160,83,374,340,280,63,394)(36,395,64,281,341,375,84,161,46,187)(49,456,410,228,406,93,424,230,315,257)(50,258,316,231,425,94,407,217,411,445)(51,446,412,218,408,95,426,232,317,259)(52,260,318,233,427,96,397,219,413,447)(53,448,414,220,398,85,428,234,319,261)(54,262,320,235,429,86,399,221,415,449)(55,450,416,222,400,87,430,236,321,263)(56,264,322,237,431,88,401,223,417,451)(57,452,418,224,402,89,432,238,323,253)(58,254,324,239,421,90,403,225,419,453)(59,454,420,226,404,91,422,240,313,255)(60,256,314,229,423,92,405,227,409,455)(97,204,126,466,266,480,439,149,137,170)(98,171,138,150,440,469,267,467,127,193)(99,194,128,468,268,470,441,151,139,172)(100,173,140,152,442,471,269,457,129,195)(101,196,130,458,270,472,443,153,141,174)(102,175,142,154,444,473,271,459,131,197)(103,198,132,460,272,474,433,155,143,176)(104,177,144,156,434,475,273,461,121,199)(105,200,122,462,274,476,435,145,133,178)(106,179,134,146,436,477,275,463,123,201)(107,202,124,464,276,478,437,147,135,180)(108,169,136,148,438,479,265,465,125,203), (1,427,242,447)(2,428,243,448)(3,429,244,449)(4,430,245,450)(5,431,246,451)(6,432,247,452)(7,421,248,453)(8,422,249,454)(9,423,250,455)(10,424,251,456)(11,425,252,445)(12,426,241,446)(13,58,352,90)(14,59,353,91)(15,60,354,92)(16,49,355,93)(17,50,356,94)(18,51,357,95)(19,52,358,96)(20,53,359,85)(21,54,360,86)(22,55,349,87)(23,56,350,88)(24,57,351,89)(25,476,376,105)(26,477,377,106)(27,478,378,107)(28,479,379,108)(29,480,380,97)(30,469,381,98)(31,470,382,99)(32,471,383,100)(33,472,384,101)(34,473,373,102)(35,474,374,103)(36,475,375,104)(37,464,284,135)(38,465,285,136)(39,466,286,137)(40,467,287,138)(41,468,288,139)(42,457,277,140)(43,458,278,141)(44,459,279,142)(45,460,280,143)(46,461,281,144)(47,462,282,133)(48,463,283,134)(61,153,158,130)(62,154,159,131)(63,155,160,132)(64,156,161,121)(65,145,162,122)(66,146,163,123)(67,147,164,124)(68,148,165,125)(69,149,166,126)(70,150,167,127)(71,151,168,128)(72,152,157,129)(73,200,396,435)(74,201,385,436)(75,202,386,437)(76,203,387,438)(77,204,388,439)(78,193,389,440)(79,194,390,441)(80,195,391,442)(81,196,392,443)(82,197,393,444)(83,198,394,433)(84,199,395,434)(109,321,296,222)(110,322,297,223)(111,323,298,224)(112,324,299,225)(113,313,300,226)(114,314,289,227)(115,315,290,228)(116,316,291,217)(117,317,292,218)(118,318,293,219)(119,319,294,220)(120,320,295,221)(169,191,265,345)(170,192,266,346)(171,181,267,347)(172,182,268,348)(173,183,269,337)(174,184,270,338)(175,185,271,339)(176,186,272,340)(177,187,273,341)(178,188,274,342)(179,189,275,343)(180,190,276,344)(205,420,330,240)(206,409,331,229)(207,410,332,230)(208,411,333,231)(209,412,334,232)(210,413,335,233)(211,414,336,234)(212,415,325,235)(213,416,326,236)(214,417,327,237)(215,418,328,238)(216,419,329,239)(253,309,402,371)(254,310,403,372)(255,311,404,361)(256,312,405,362)(257,301,406,363)(258,302,407,364)(259,303,408,365)(260,304,397,366)(261,305,398,367)(262,306,399,368)(263,307,400,369)(264,308,401,370), (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), (1,388,7,394)(2,76,8,82)(3,386,9,392)(4,74,10,80)(5,396,11,390)(6,84,12,78)(13,160,19,166)(14,62,20,68)(15,158,21,164)(16,72,22,66)(17,168,23,162)(18,70,24,64)(25,333,31,327)(26,207,32,213)(27,331,33,325)(28,205,34,211)(29,329,35,335)(30,215,36,209)(37,362,43,368)(38,311,44,305)(39,372,45,366)(40,309,46,303)(41,370,47,364)(42,307,48,301)(49,129,55,123)(50,151,56,145)(51,127,57,121)(52,149,58,155)(53,125,59,131)(54,147,60,153)(61,360,67,354)(63,358,69,352)(65,356,71,350)(73,252,79,246)(75,250,81,244)(77,248,83,242)(85,148,91,154)(86,124,92,130)(87,146,93,152)(88,122,94,128)(89,156,95,150)(90,132,96,126)(97,239,103,233)(98,418,104,412)(99,237,105,231)(100,416,106,410)(101,235,107,229)(102,414,108,420)(109,343,115,337)(110,188,116,182)(111,341,117,347)(112,186,118,192)(113,339,119,345)(114,184,120,190)(133,258,139,264)(134,406,140,400)(135,256,141,262)(136,404,142,398)(137,254,143,260)(138,402,144,408)(157,349,163,355)(159,359,165,353)(161,357,167,351)(169,226,175,220)(170,324,176,318)(171,224,177,218)(172,322,178,316)(173,222,179,228)(174,320,180,314)(181,298,187,292)(183,296,189,290)(185,294,191,300)(193,452,199,446)(194,431,200,425)(195,450,201,456)(196,429,202,423)(197,448,203,454)(198,427,204,421)(206,384,212,378)(208,382,214,376)(210,380,216,374)(217,268,223,274)(219,266,225,272)(221,276,227,270)(230,471,236,477)(232,469,238,475)(234,479,240,473)(241,389,247,395)(243,387,249,393)(245,385,251,391)(253,461,259,467)(255,459,261,465)(257,457,263,463)(265,313,271,319)(267,323,273,317)(269,321,275,315)(277,369,283,363)(278,306,284,312)(279,367,285,361)(280,304,286,310)(281,365,287,371)(282,302,288,308)(289,338,295,344)(291,348,297,342)(293,346,299,340)(326,377,332,383)(328,375,334,381)(330,373,336,379)(397,466,403,460)(399,464,405,458)(401,462,407,468)(409,472,415,478)(411,470,417,476)(413,480,419,474)(422,444,428,438)(424,442,430,436)(426,440,432,434)(433,447,439,453)(435,445,441,451)(437,455,443,449)>;
G:=Group( (1,335,118,366,19,242,210,293,304,358)(2,359,305,294,211,243,20,367,119,336)(3,325,120,368,21,244,212,295,306,360)(4,349,307,296,213,245,22,369,109,326)(5,327,110,370,23,246,214,297,308,350)(6,351,309,298,215,247,24,371,111,328)(7,329,112,372,13,248,216,299,310,352)(8,353,311,300,205,249,14,361,113,330)(9,331,114,362,15,250,206,289,312,354)(10,355,301,290,207,251,16,363,115,332)(11,333,116,364,17,252,208,291,302,356)(12,357,303,292,209,241,18,365,117,334)(25,188,47,162,73,376,342,282,65,396)(26,385,66,283,343,377,74,163,48,189)(27,190,37,164,75,378,344,284,67,386)(28,387,68,285,345,379,76,165,38,191)(29,192,39,166,77,380,346,286,69,388)(30,389,70,287,347,381,78,167,40,181)(31,182,41,168,79,382,348,288,71,390)(32,391,72,277,337,383,80,157,42,183)(33,184,43,158,81,384,338,278,61,392)(34,393,62,279,339,373,82,159,44,185)(35,186,45,160,83,374,340,280,63,394)(36,395,64,281,341,375,84,161,46,187)(49,456,410,228,406,93,424,230,315,257)(50,258,316,231,425,94,407,217,411,445)(51,446,412,218,408,95,426,232,317,259)(52,260,318,233,427,96,397,219,413,447)(53,448,414,220,398,85,428,234,319,261)(54,262,320,235,429,86,399,221,415,449)(55,450,416,222,400,87,430,236,321,263)(56,264,322,237,431,88,401,223,417,451)(57,452,418,224,402,89,432,238,323,253)(58,254,324,239,421,90,403,225,419,453)(59,454,420,226,404,91,422,240,313,255)(60,256,314,229,423,92,405,227,409,455)(97,204,126,466,266,480,439,149,137,170)(98,171,138,150,440,469,267,467,127,193)(99,194,128,468,268,470,441,151,139,172)(100,173,140,152,442,471,269,457,129,195)(101,196,130,458,270,472,443,153,141,174)(102,175,142,154,444,473,271,459,131,197)(103,198,132,460,272,474,433,155,143,176)(104,177,144,156,434,475,273,461,121,199)(105,200,122,462,274,476,435,145,133,178)(106,179,134,146,436,477,275,463,123,201)(107,202,124,464,276,478,437,147,135,180)(108,169,136,148,438,479,265,465,125,203), (1,427,242,447)(2,428,243,448)(3,429,244,449)(4,430,245,450)(5,431,246,451)(6,432,247,452)(7,421,248,453)(8,422,249,454)(9,423,250,455)(10,424,251,456)(11,425,252,445)(12,426,241,446)(13,58,352,90)(14,59,353,91)(15,60,354,92)(16,49,355,93)(17,50,356,94)(18,51,357,95)(19,52,358,96)(20,53,359,85)(21,54,360,86)(22,55,349,87)(23,56,350,88)(24,57,351,89)(25,476,376,105)(26,477,377,106)(27,478,378,107)(28,479,379,108)(29,480,380,97)(30,469,381,98)(31,470,382,99)(32,471,383,100)(33,472,384,101)(34,473,373,102)(35,474,374,103)(36,475,375,104)(37,464,284,135)(38,465,285,136)(39,466,286,137)(40,467,287,138)(41,468,288,139)(42,457,277,140)(43,458,278,141)(44,459,279,142)(45,460,280,143)(46,461,281,144)(47,462,282,133)(48,463,283,134)(61,153,158,130)(62,154,159,131)(63,155,160,132)(64,156,161,121)(65,145,162,122)(66,146,163,123)(67,147,164,124)(68,148,165,125)(69,149,166,126)(70,150,167,127)(71,151,168,128)(72,152,157,129)(73,200,396,435)(74,201,385,436)(75,202,386,437)(76,203,387,438)(77,204,388,439)(78,193,389,440)(79,194,390,441)(80,195,391,442)(81,196,392,443)(82,197,393,444)(83,198,394,433)(84,199,395,434)(109,321,296,222)(110,322,297,223)(111,323,298,224)(112,324,299,225)(113,313,300,226)(114,314,289,227)(115,315,290,228)(116,316,291,217)(117,317,292,218)(118,318,293,219)(119,319,294,220)(120,320,295,221)(169,191,265,345)(170,192,266,346)(171,181,267,347)(172,182,268,348)(173,183,269,337)(174,184,270,338)(175,185,271,339)(176,186,272,340)(177,187,273,341)(178,188,274,342)(179,189,275,343)(180,190,276,344)(205,420,330,240)(206,409,331,229)(207,410,332,230)(208,411,333,231)(209,412,334,232)(210,413,335,233)(211,414,336,234)(212,415,325,235)(213,416,326,236)(214,417,327,237)(215,418,328,238)(216,419,329,239)(253,309,402,371)(254,310,403,372)(255,311,404,361)(256,312,405,362)(257,301,406,363)(258,302,407,364)(259,303,408,365)(260,304,397,366)(261,305,398,367)(262,306,399,368)(263,307,400,369)(264,308,401,370), (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), (1,388,7,394)(2,76,8,82)(3,386,9,392)(4,74,10,80)(5,396,11,390)(6,84,12,78)(13,160,19,166)(14,62,20,68)(15,158,21,164)(16,72,22,66)(17,168,23,162)(18,70,24,64)(25,333,31,327)(26,207,32,213)(27,331,33,325)(28,205,34,211)(29,329,35,335)(30,215,36,209)(37,362,43,368)(38,311,44,305)(39,372,45,366)(40,309,46,303)(41,370,47,364)(42,307,48,301)(49,129,55,123)(50,151,56,145)(51,127,57,121)(52,149,58,155)(53,125,59,131)(54,147,60,153)(61,360,67,354)(63,358,69,352)(65,356,71,350)(73,252,79,246)(75,250,81,244)(77,248,83,242)(85,148,91,154)(86,124,92,130)(87,146,93,152)(88,122,94,128)(89,156,95,150)(90,132,96,126)(97,239,103,233)(98,418,104,412)(99,237,105,231)(100,416,106,410)(101,235,107,229)(102,414,108,420)(109,343,115,337)(110,188,116,182)(111,341,117,347)(112,186,118,192)(113,339,119,345)(114,184,120,190)(133,258,139,264)(134,406,140,400)(135,256,141,262)(136,404,142,398)(137,254,143,260)(138,402,144,408)(157,349,163,355)(159,359,165,353)(161,357,167,351)(169,226,175,220)(170,324,176,318)(171,224,177,218)(172,322,178,316)(173,222,179,228)(174,320,180,314)(181,298,187,292)(183,296,189,290)(185,294,191,300)(193,452,199,446)(194,431,200,425)(195,450,201,456)(196,429,202,423)(197,448,203,454)(198,427,204,421)(206,384,212,378)(208,382,214,376)(210,380,216,374)(217,268,223,274)(219,266,225,272)(221,276,227,270)(230,471,236,477)(232,469,238,475)(234,479,240,473)(241,389,247,395)(243,387,249,393)(245,385,251,391)(253,461,259,467)(255,459,261,465)(257,457,263,463)(265,313,271,319)(267,323,273,317)(269,321,275,315)(277,369,283,363)(278,306,284,312)(279,367,285,361)(280,304,286,310)(281,365,287,371)(282,302,288,308)(289,338,295,344)(291,348,297,342)(293,346,299,340)(326,377,332,383)(328,375,334,381)(330,373,336,379)(397,466,403,460)(399,464,405,458)(401,462,407,468)(409,472,415,478)(411,470,417,476)(413,480,419,474)(422,444,428,438)(424,442,430,436)(426,440,432,434)(433,447,439,453)(435,445,441,451)(437,455,443,449) );
G=PermutationGroup([[(1,335,118,366,19,242,210,293,304,358),(2,359,305,294,211,243,20,367,119,336),(3,325,120,368,21,244,212,295,306,360),(4,349,307,296,213,245,22,369,109,326),(5,327,110,370,23,246,214,297,308,350),(6,351,309,298,215,247,24,371,111,328),(7,329,112,372,13,248,216,299,310,352),(8,353,311,300,205,249,14,361,113,330),(9,331,114,362,15,250,206,289,312,354),(10,355,301,290,207,251,16,363,115,332),(11,333,116,364,17,252,208,291,302,356),(12,357,303,292,209,241,18,365,117,334),(25,188,47,162,73,376,342,282,65,396),(26,385,66,283,343,377,74,163,48,189),(27,190,37,164,75,378,344,284,67,386),(28,387,68,285,345,379,76,165,38,191),(29,192,39,166,77,380,346,286,69,388),(30,389,70,287,347,381,78,167,40,181),(31,182,41,168,79,382,348,288,71,390),(32,391,72,277,337,383,80,157,42,183),(33,184,43,158,81,384,338,278,61,392),(34,393,62,279,339,373,82,159,44,185),(35,186,45,160,83,374,340,280,63,394),(36,395,64,281,341,375,84,161,46,187),(49,456,410,228,406,93,424,230,315,257),(50,258,316,231,425,94,407,217,411,445),(51,446,412,218,408,95,426,232,317,259),(52,260,318,233,427,96,397,219,413,447),(53,448,414,220,398,85,428,234,319,261),(54,262,320,235,429,86,399,221,415,449),(55,450,416,222,400,87,430,236,321,263),(56,264,322,237,431,88,401,223,417,451),(57,452,418,224,402,89,432,238,323,253),(58,254,324,239,421,90,403,225,419,453),(59,454,420,226,404,91,422,240,313,255),(60,256,314,229,423,92,405,227,409,455),(97,204,126,466,266,480,439,149,137,170),(98,171,138,150,440,469,267,467,127,193),(99,194,128,468,268,470,441,151,139,172),(100,173,140,152,442,471,269,457,129,195),(101,196,130,458,270,472,443,153,141,174),(102,175,142,154,444,473,271,459,131,197),(103,198,132,460,272,474,433,155,143,176),(104,177,144,156,434,475,273,461,121,199),(105,200,122,462,274,476,435,145,133,178),(106,179,134,146,436,477,275,463,123,201),(107,202,124,464,276,478,437,147,135,180),(108,169,136,148,438,479,265,465,125,203)], [(1,427,242,447),(2,428,243,448),(3,429,244,449),(4,430,245,450),(5,431,246,451),(6,432,247,452),(7,421,248,453),(8,422,249,454),(9,423,250,455),(10,424,251,456),(11,425,252,445),(12,426,241,446),(13,58,352,90),(14,59,353,91),(15,60,354,92),(16,49,355,93),(17,50,356,94),(18,51,357,95),(19,52,358,96),(20,53,359,85),(21,54,360,86),(22,55,349,87),(23,56,350,88),(24,57,351,89),(25,476,376,105),(26,477,377,106),(27,478,378,107),(28,479,379,108),(29,480,380,97),(30,469,381,98),(31,470,382,99),(32,471,383,100),(33,472,384,101),(34,473,373,102),(35,474,374,103),(36,475,375,104),(37,464,284,135),(38,465,285,136),(39,466,286,137),(40,467,287,138),(41,468,288,139),(42,457,277,140),(43,458,278,141),(44,459,279,142),(45,460,280,143),(46,461,281,144),(47,462,282,133),(48,463,283,134),(61,153,158,130),(62,154,159,131),(63,155,160,132),(64,156,161,121),(65,145,162,122),(66,146,163,123),(67,147,164,124),(68,148,165,125),(69,149,166,126),(70,150,167,127),(71,151,168,128),(72,152,157,129),(73,200,396,435),(74,201,385,436),(75,202,386,437),(76,203,387,438),(77,204,388,439),(78,193,389,440),(79,194,390,441),(80,195,391,442),(81,196,392,443),(82,197,393,444),(83,198,394,433),(84,199,395,434),(109,321,296,222),(110,322,297,223),(111,323,298,224),(112,324,299,225),(113,313,300,226),(114,314,289,227),(115,315,290,228),(116,316,291,217),(117,317,292,218),(118,318,293,219),(119,319,294,220),(120,320,295,221),(169,191,265,345),(170,192,266,346),(171,181,267,347),(172,182,268,348),(173,183,269,337),(174,184,270,338),(175,185,271,339),(176,186,272,340),(177,187,273,341),(178,188,274,342),(179,189,275,343),(180,190,276,344),(205,420,330,240),(206,409,331,229),(207,410,332,230),(208,411,333,231),(209,412,334,232),(210,413,335,233),(211,414,336,234),(212,415,325,235),(213,416,326,236),(214,417,327,237),(215,418,328,238),(216,419,329,239),(253,309,402,371),(254,310,403,372),(255,311,404,361),(256,312,405,362),(257,301,406,363),(258,302,407,364),(259,303,408,365),(260,304,397,366),(261,305,398,367),(262,306,399,368),(263,307,400,369),(264,308,401,370)], [(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)], [(1,388,7,394),(2,76,8,82),(3,386,9,392),(4,74,10,80),(5,396,11,390),(6,84,12,78),(13,160,19,166),(14,62,20,68),(15,158,21,164),(16,72,22,66),(17,168,23,162),(18,70,24,64),(25,333,31,327),(26,207,32,213),(27,331,33,325),(28,205,34,211),(29,329,35,335),(30,215,36,209),(37,362,43,368),(38,311,44,305),(39,372,45,366),(40,309,46,303),(41,370,47,364),(42,307,48,301),(49,129,55,123),(50,151,56,145),(51,127,57,121),(52,149,58,155),(53,125,59,131),(54,147,60,153),(61,360,67,354),(63,358,69,352),(65,356,71,350),(73,252,79,246),(75,250,81,244),(77,248,83,242),(85,148,91,154),(86,124,92,130),(87,146,93,152),(88,122,94,128),(89,156,95,150),(90,132,96,126),(97,239,103,233),(98,418,104,412),(99,237,105,231),(100,416,106,410),(101,235,107,229),(102,414,108,420),(109,343,115,337),(110,188,116,182),(111,341,117,347),(112,186,118,192),(113,339,119,345),(114,184,120,190),(133,258,139,264),(134,406,140,400),(135,256,141,262),(136,404,142,398),(137,254,143,260),(138,402,144,408),(157,349,163,355),(159,359,165,353),(161,357,167,351),(169,226,175,220),(170,324,176,318),(171,224,177,218),(172,322,178,316),(173,222,179,228),(174,320,180,314),(181,298,187,292),(183,296,189,290),(185,294,191,300),(193,452,199,446),(194,431,200,425),(195,450,201,456),(196,429,202,423),(197,448,203,454),(198,427,204,421),(206,384,212,378),(208,382,214,376),(210,380,216,374),(217,268,223,274),(219,266,225,272),(221,276,227,270),(230,471,236,477),(232,469,238,475),(234,479,240,473),(241,389,247,395),(243,387,249,393),(245,385,251,391),(253,461,259,467),(255,459,261,465),(257,457,263,463),(265,313,271,319),(267,323,273,317),(269,321,275,315),(277,369,283,363),(278,306,284,312),(279,367,285,361),(280,304,286,310),(281,365,287,371),(282,302,288,308),(289,338,295,344),(291,348,297,342),(293,346,299,340),(326,377,332,383),(328,375,334,381),(330,373,336,379),(397,466,403,460),(399,464,405,458),(401,462,407,468),(409,472,415,478),(411,470,417,476),(413,480,419,474),(422,444,428,438),(424,442,430,436),(426,440,432,434),(433,447,439,453),(435,445,441,451),(437,455,443,449)]])
66 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 | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 30A | ··· | 30F | 60A | ··· | 60H |
| 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 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 12 | 12 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | - | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | Dic6 | C4○D12 | C4○D20 | S3×D5 | D4⋊2D5 | Q8×D5 | C2×S3×D5 | D5×Dic6 | D6.D10 | Dic3.D10 |
| kernel | Dic5.7Dic6 | C30.Q8 | Dic15⋊5C4 | C6.Dic10 | C12×Dic5 | C5×Dic3⋊C4 | C30.4Q8 | C4×Dic5 | C3×Dic5 | Dic3⋊C4 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | Dic5 | C10 | C6 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic5.7Dic6 ►in GL6(𝔽61)
| 17 | 60 | 0 | 0 | 0 | 0 |
| 45 | 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 | 60 |
| 58 | 52 | 0 | 0 | 0 | 0 |
| 35 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 15 |
| 0 | 0 | 0 | 0 | 46 | 38 |
| 33 | 38 | 0 | 0 | 0 | 0 |
| 42 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 20 | 0 | 0 |
| 0 | 0 | 2 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 52 |
| 0 | 0 | 0 | 0 | 9 | 43 |
| 14 | 44 | 0 | 0 | 0 | 0 |
| 33 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 51 | 0 | 0 |
| 0 | 0 | 38 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 19 |
| 0 | 0 | 0 | 0 | 49 | 31 |
G:=sub<GL(6,GF(61))| [17,45,0,0,0,0,60,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[58,35,0,0,0,0,52,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,23,46,0,0,0,0,15,38],[33,42,0,0,0,0,38,28,0,0,0,0,0,0,9,2,0,0,0,0,20,52,0,0,0,0,0,0,52,9,0,0,0,0,52,43],[14,33,0,0,0,0,44,47,0,0,0,0,0,0,47,38,0,0,0,0,51,14,0,0,0,0,0,0,30,49,0,0,0,0,19,31] >;
Dic5.7Dic6 in GAP, Magma, Sage, TeX
{\rm Dic}_5._7{\rm Dic}_6 % in TeX
G:=Group("Dic5.7Dic6"); // GroupNames label
G:=SmallGroup(480,454);
// by ID
G=gap.SmallGroup(480,454);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=c^6,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=a^5*c^-1>;
// generators/relations