metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic5.2Dic6, C6.25(Q8×D5), (C2×C12).4D10, C4⋊Dic3.4D5, C30.12(C2×Q8), C2.9(D5×Dic6), C30.6(C4○D4), (C2×C20).218D6, (C3×Dic5).2Q8, C10.7(C2×Dic6), C5⋊3(C4.Dic6), C15⋊2(C42.C2), C6.61(C4○D20), C6.3(D4⋊2D5), (C2×C30).25C23, (C2×Dic5).87D6, (C2×Dic3).2D10, C10.D4.8S3, C6.Dic10.5C2, C30.Q8.7C2, C2.7(D12⋊D5), (C2×C60).311C22, C3⋊3(Dic5.Q8), C30.4Q8.10C2, C10.36(D4⋊2S3), C10.21(Q8⋊3S3), (Dic3×Dic5).12C2, C2.8(Dic5.D6), (C6×Dic5).10C22, (C2×Dic15).32C22, (C10×Dic3).11C22, (C2×C4).24(S3×D5), C22.118(C2×S3×D5), (C5×C4⋊Dic3).16C2, (C2×C6).37(C22×D5), (C2×C10).37(C22×S3), (C3×C10.D4).10C2, SmallGroup(480,411)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic5.2Dic6
G = < a,b,c,d | a10=c12=1, b2=a5, d2=a5c6, bab-1=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c-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, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C5×C4⋊C4, C4.Dic6, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, Dic5.Q8, Dic3×Dic5, C30.Q8, C6.Dic10, C3×C10.D4, C5×C4⋊Dic3, C30.4Q8, Dic5.2Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, Dic6, C22×S3, C42.C2, C22×D5, C2×Dic6, D4⋊2S3, Q8⋊3S3, S3×D5, C4○D20, D4⋊2D5, Q8×D5, C4.Dic6, C2×S3×D5, Dic5.Q8, D5×Dic6, D12⋊D5, Dic5.D6, Dic5.2Dic6
►Regular action on 480 points
Generators in S480
(1 201 281 298 327 405 381 54 179 144)(2 202 282 299 328 406 382 55 180 133)(3 203 283 300 329 407 383 56 169 134)(4 204 284 289 330 408 384 57 170 135)(5 193 285 290 331 397 373 58 171 136)(6 194 286 291 332 398 374 59 172 137)(7 195 287 292 333 399 375 60 173 138)(8 196 288 293 334 400 376 49 174 139)(9 197 277 294 335 401 377 50 175 140)(10 198 278 295 336 402 378 51 176 141)(11 199 279 296 325 403 379 52 177 142)(12 200 280 297 326 404 380 53 178 143)(13 460 436 254 414 320 104 125 450 274)(14 461 437 255 415 321 105 126 451 275)(15 462 438 256 416 322 106 127 452 276)(16 463 439 257 417 323 107 128 453 265)(17 464 440 258 418 324 108 129 454 266)(18 465 441 259 419 313 97 130 455 267)(19 466 442 260 420 314 98 131 456 268)(20 467 443 261 409 315 99 132 445 269)(21 468 444 262 410 316 100 121 446 270)(22 457 433 263 411 317 101 122 447 271)(23 458 434 264 412 318 102 123 448 272)(24 459 435 253 413 319 103 124 449 273)(25 44 222 68 87 161 309 206 395 428)(26 45 223 69 88 162 310 207 396 429)(27 46 224 70 89 163 311 208 385 430)(28 47 225 71 90 164 312 209 386 431)(29 48 226 72 91 165 301 210 387 432)(30 37 227 61 92 166 302 211 388 421)(31 38 228 62 93 167 303 212 389 422)(32 39 217 63 94 168 304 213 390 423)(33 40 218 64 95 157 305 214 391 424)(34 41 219 65 96 158 306 215 392 425)(35 42 220 66 85 159 307 216 393 426)(36 43 221 67 86 160 308 205 394 427)(73 243 235 114 350 470 337 182 367 156)(74 244 236 115 351 471 338 183 368 145)(75 245 237 116 352 472 339 184 369 146)(76 246 238 117 353 473 340 185 370 147)(77 247 239 118 354 474 341 186 371 148)(78 248 240 119 355 475 342 187 372 149)(79 249 229 120 356 476 343 188 361 150)(80 250 230 109 357 477 344 189 362 151)(81 251 231 110 358 478 345 190 363 152)(82 252 232 111 359 479 346 191 364 153)(83 241 233 112 360 480 347 192 365 154)(84 242 234 113 349 469 348 181 366 155)
(1 103 405 459)(2 460 406 104)(3 105 407 461)(4 462 408 106)(5 107 397 463)(6 464 398 108)(7 97 399 465)(8 466 400 98)(9 99 401 467)(10 468 402 100)(11 101 403 457)(12 458 404 102)(13 382 320 202)(14 203 321 383)(15 384 322 204)(16 193 323 373)(17 374 324 194)(18 195 313 375)(19 376 314 196)(20 197 315 377)(21 378 316 198)(22 199 317 379)(23 380 318 200)(24 201 319 381)(25 351 161 145)(26 146 162 352)(27 353 163 147)(28 148 164 354)(29 355 165 149)(30 150 166 356)(31 357 167 151)(32 152 168 358)(33 359 157 153)(34 154 158 360)(35 349 159 155)(36 156 160 350)(37 361 302 120)(38 109 303 362)(39 363 304 110)(40 111 305 364)(41 365 306 112)(42 113 307 366)(43 367 308 114)(44 115 309 368)(45 369 310 116)(46 117 311 370)(47 371 312 118)(48 119 301 372)(49 420 288 268)(50 269 277 409)(51 410 278 270)(52 271 279 411)(53 412 280 272)(54 273 281 413)(55 414 282 274)(56 275 283 415)(57 416 284 276)(58 265 285 417)(59 418 286 266)(60 267 287 419)(61 343 388 249)(62 250 389 344)(63 345 390 251)(64 252 391 346)(65 347 392 241)(66 242 393 348)(67 337 394 243)(68 244 395 338)(69 339 396 245)(70 246 385 340)(71 341 386 247)(72 248 387 342)(73 86 470 427)(74 428 471 87)(75 88 472 429)(76 430 473 89)(77 90 474 431)(78 432 475 91)(79 92 476 421)(80 422 477 93)(81 94 478 423)(82 424 479 95)(83 96 480 425)(84 426 469 85)(121 141 444 336)(122 325 433 142)(123 143 434 326)(124 327 435 144)(125 133 436 328)(126 329 437 134)(127 135 438 330)(128 331 439 136)(129 137 440 332)(130 333 441 138)(131 139 442 334)(132 335 443 140)(169 451 300 255)(170 256 289 452)(171 453 290 257)(172 258 291 454)(173 455 292 259)(174 260 293 456)(175 445 294 261)(176 262 295 446)(177 447 296 263)(178 264 297 448)(179 449 298 253)(180 254 299 450)(181 220 234 216)(182 205 235 221)(183 222 236 206)(184 207 237 223)(185 224 238 208)(186 209 239 225)(187 226 240 210)(188 211 229 227)(189 228 230 212)(190 213 231 217)(191 218 232 214)(192 215 233 219)
(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 89 399 424)(2 88 400 423)(3 87 401 422)(4 86 402 421)(5 85 403 432)(6 96 404 431)(7 95 405 430)(8 94 406 429)(9 93 407 428)(10 92 408 427)(11 91 397 426)(12 90 398 425)(13 352 314 152)(14 351 315 151)(15 350 316 150)(16 349 317 149)(17 360 318 148)(18 359 319 147)(19 358 320 146)(20 357 321 145)(21 356 322 156)(22 355 323 155)(23 354 324 154)(24 353 313 153)(25 197 167 383)(26 196 168 382)(27 195 157 381)(28 194 158 380)(29 193 159 379)(30 204 160 378)(31 203 161 377)(32 202 162 376)(33 201 163 375)(34 200 164 374)(35 199 165 373)(36 198 166 384)(37 284 308 51)(38 283 309 50)(39 282 310 49)(40 281 311 60)(41 280 312 59)(42 279 301 58)(43 278 302 57)(44 277 303 56)(45 288 304 55)(46 287 305 54)(47 286 306 53)(48 285 307 52)(61 330 394 141)(62 329 395 140)(63 328 396 139)(64 327 385 138)(65 326 386 137)(66 325 387 136)(67 336 388 135)(68 335 389 134)(69 334 390 133)(70 333 391 144)(71 332 392 143)(72 331 393 142)(73 468 476 106)(74 467 477 105)(75 466 478 104)(76 465 479 103)(77 464 480 102)(78 463 469 101)(79 462 470 100)(80 461 471 99)(81 460 472 98)(82 459 473 97)(83 458 474 108)(84 457 475 107)(109 415 368 269)(110 414 369 268)(111 413 370 267)(112 412 371 266)(113 411 372 265)(114 410 361 276)(115 409 362 275)(116 420 363 274)(117 419 364 273)(118 418 365 272)(119 417 366 271)(120 416 367 270)(121 249 438 337)(122 248 439 348)(123 247 440 347)(124 246 441 346)(125 245 442 345)(126 244 443 344)(127 243 444 343)(128 242 433 342)(129 241 434 341)(130 252 435 340)(131 251 436 339)(132 250 437 338)(169 222 294 212)(170 221 295 211)(171 220 296 210)(172 219 297 209)(173 218 298 208)(174 217 299 207)(175 228 300 206)(176 227 289 205)(177 226 290 216)(178 225 291 215)(179 224 292 214)(180 223 293 213)(181 447 240 257)(182 446 229 256)(183 445 230 255)(184 456 231 254)(185 455 232 253)(186 454 233 264)(187 453 234 263)(188 452 235 262)(189 451 236 261)(190 450 237 260)(191 449 238 259)(192 448 239 258)
G:=sub<Sym(480)| (1,201,281,298,327,405,381,54,179,144)(2,202,282,299,328,406,382,55,180,133)(3,203,283,300,329,407,383,56,169,134)(4,204,284,289,330,408,384,57,170,135)(5,193,285,290,331,397,373,58,171,136)(6,194,286,291,332,398,374,59,172,137)(7,195,287,292,333,399,375,60,173,138)(8,196,288,293,334,400,376,49,174,139)(9,197,277,294,335,401,377,50,175,140)(10,198,278,295,336,402,378,51,176,141)(11,199,279,296,325,403,379,52,177,142)(12,200,280,297,326,404,380,53,178,143)(13,460,436,254,414,320,104,125,450,274)(14,461,437,255,415,321,105,126,451,275)(15,462,438,256,416,322,106,127,452,276)(16,463,439,257,417,323,107,128,453,265)(17,464,440,258,418,324,108,129,454,266)(18,465,441,259,419,313,97,130,455,267)(19,466,442,260,420,314,98,131,456,268)(20,467,443,261,409,315,99,132,445,269)(21,468,444,262,410,316,100,121,446,270)(22,457,433,263,411,317,101,122,447,271)(23,458,434,264,412,318,102,123,448,272)(24,459,435,253,413,319,103,124,449,273)(25,44,222,68,87,161,309,206,395,428)(26,45,223,69,88,162,310,207,396,429)(27,46,224,70,89,163,311,208,385,430)(28,47,225,71,90,164,312,209,386,431)(29,48,226,72,91,165,301,210,387,432)(30,37,227,61,92,166,302,211,388,421)(31,38,228,62,93,167,303,212,389,422)(32,39,217,63,94,168,304,213,390,423)(33,40,218,64,95,157,305,214,391,424)(34,41,219,65,96,158,306,215,392,425)(35,42,220,66,85,159,307,216,393,426)(36,43,221,67,86,160,308,205,394,427)(73,243,235,114,350,470,337,182,367,156)(74,244,236,115,351,471,338,183,368,145)(75,245,237,116,352,472,339,184,369,146)(76,246,238,117,353,473,340,185,370,147)(77,247,239,118,354,474,341,186,371,148)(78,248,240,119,355,475,342,187,372,149)(79,249,229,120,356,476,343,188,361,150)(80,250,230,109,357,477,344,189,362,151)(81,251,231,110,358,478,345,190,363,152)(82,252,232,111,359,479,346,191,364,153)(83,241,233,112,360,480,347,192,365,154)(84,242,234,113,349,469,348,181,366,155), (1,103,405,459)(2,460,406,104)(3,105,407,461)(4,462,408,106)(5,107,397,463)(6,464,398,108)(7,97,399,465)(8,466,400,98)(9,99,401,467)(10,468,402,100)(11,101,403,457)(12,458,404,102)(13,382,320,202)(14,203,321,383)(15,384,322,204)(16,193,323,373)(17,374,324,194)(18,195,313,375)(19,376,314,196)(20,197,315,377)(21,378,316,198)(22,199,317,379)(23,380,318,200)(24,201,319,381)(25,351,161,145)(26,146,162,352)(27,353,163,147)(28,148,164,354)(29,355,165,149)(30,150,166,356)(31,357,167,151)(32,152,168,358)(33,359,157,153)(34,154,158,360)(35,349,159,155)(36,156,160,350)(37,361,302,120)(38,109,303,362)(39,363,304,110)(40,111,305,364)(41,365,306,112)(42,113,307,366)(43,367,308,114)(44,115,309,368)(45,369,310,116)(46,117,311,370)(47,371,312,118)(48,119,301,372)(49,420,288,268)(50,269,277,409)(51,410,278,270)(52,271,279,411)(53,412,280,272)(54,273,281,413)(55,414,282,274)(56,275,283,415)(57,416,284,276)(58,265,285,417)(59,418,286,266)(60,267,287,419)(61,343,388,249)(62,250,389,344)(63,345,390,251)(64,252,391,346)(65,347,392,241)(66,242,393,348)(67,337,394,243)(68,244,395,338)(69,339,396,245)(70,246,385,340)(71,341,386,247)(72,248,387,342)(73,86,470,427)(74,428,471,87)(75,88,472,429)(76,430,473,89)(77,90,474,431)(78,432,475,91)(79,92,476,421)(80,422,477,93)(81,94,478,423)(82,424,479,95)(83,96,480,425)(84,426,469,85)(121,141,444,336)(122,325,433,142)(123,143,434,326)(124,327,435,144)(125,133,436,328)(126,329,437,134)(127,135,438,330)(128,331,439,136)(129,137,440,332)(130,333,441,138)(131,139,442,334)(132,335,443,140)(169,451,300,255)(170,256,289,452)(171,453,290,257)(172,258,291,454)(173,455,292,259)(174,260,293,456)(175,445,294,261)(176,262,295,446)(177,447,296,263)(178,264,297,448)(179,449,298,253)(180,254,299,450)(181,220,234,216)(182,205,235,221)(183,222,236,206)(184,207,237,223)(185,224,238,208)(186,209,239,225)(187,226,240,210)(188,211,229,227)(189,228,230,212)(190,213,231,217)(191,218,232,214)(192,215,233,219), (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), (1,89,399,424)(2,88,400,423)(3,87,401,422)(4,86,402,421)(5,85,403,432)(6,96,404,431)(7,95,405,430)(8,94,406,429)(9,93,407,428)(10,92,408,427)(11,91,397,426)(12,90,398,425)(13,352,314,152)(14,351,315,151)(15,350,316,150)(16,349,317,149)(17,360,318,148)(18,359,319,147)(19,358,320,146)(20,357,321,145)(21,356,322,156)(22,355,323,155)(23,354,324,154)(24,353,313,153)(25,197,167,383)(26,196,168,382)(27,195,157,381)(28,194,158,380)(29,193,159,379)(30,204,160,378)(31,203,161,377)(32,202,162,376)(33,201,163,375)(34,200,164,374)(35,199,165,373)(36,198,166,384)(37,284,308,51)(38,283,309,50)(39,282,310,49)(40,281,311,60)(41,280,312,59)(42,279,301,58)(43,278,302,57)(44,277,303,56)(45,288,304,55)(46,287,305,54)(47,286,306,53)(48,285,307,52)(61,330,394,141)(62,329,395,140)(63,328,396,139)(64,327,385,138)(65,326,386,137)(66,325,387,136)(67,336,388,135)(68,335,389,134)(69,334,390,133)(70,333,391,144)(71,332,392,143)(72,331,393,142)(73,468,476,106)(74,467,477,105)(75,466,478,104)(76,465,479,103)(77,464,480,102)(78,463,469,101)(79,462,470,100)(80,461,471,99)(81,460,472,98)(82,459,473,97)(83,458,474,108)(84,457,475,107)(109,415,368,269)(110,414,369,268)(111,413,370,267)(112,412,371,266)(113,411,372,265)(114,410,361,276)(115,409,362,275)(116,420,363,274)(117,419,364,273)(118,418,365,272)(119,417,366,271)(120,416,367,270)(121,249,438,337)(122,248,439,348)(123,247,440,347)(124,246,441,346)(125,245,442,345)(126,244,443,344)(127,243,444,343)(128,242,433,342)(129,241,434,341)(130,252,435,340)(131,251,436,339)(132,250,437,338)(169,222,294,212)(170,221,295,211)(171,220,296,210)(172,219,297,209)(173,218,298,208)(174,217,299,207)(175,228,300,206)(176,227,289,205)(177,226,290,216)(178,225,291,215)(179,224,292,214)(180,223,293,213)(181,447,240,257)(182,446,229,256)(183,445,230,255)(184,456,231,254)(185,455,232,253)(186,454,233,264)(187,453,234,263)(188,452,235,262)(189,451,236,261)(190,450,237,260)(191,449,238,259)(192,448,239,258)>;
G:=Group( (1,201,281,298,327,405,381,54,179,144)(2,202,282,299,328,406,382,55,180,133)(3,203,283,300,329,407,383,56,169,134)(4,204,284,289,330,408,384,57,170,135)(5,193,285,290,331,397,373,58,171,136)(6,194,286,291,332,398,374,59,172,137)(7,195,287,292,333,399,375,60,173,138)(8,196,288,293,334,400,376,49,174,139)(9,197,277,294,335,401,377,50,175,140)(10,198,278,295,336,402,378,51,176,141)(11,199,279,296,325,403,379,52,177,142)(12,200,280,297,326,404,380,53,178,143)(13,460,436,254,414,320,104,125,450,274)(14,461,437,255,415,321,105,126,451,275)(15,462,438,256,416,322,106,127,452,276)(16,463,439,257,417,323,107,128,453,265)(17,464,440,258,418,324,108,129,454,266)(18,465,441,259,419,313,97,130,455,267)(19,466,442,260,420,314,98,131,456,268)(20,467,443,261,409,315,99,132,445,269)(21,468,444,262,410,316,100,121,446,270)(22,457,433,263,411,317,101,122,447,271)(23,458,434,264,412,318,102,123,448,272)(24,459,435,253,413,319,103,124,449,273)(25,44,222,68,87,161,309,206,395,428)(26,45,223,69,88,162,310,207,396,429)(27,46,224,70,89,163,311,208,385,430)(28,47,225,71,90,164,312,209,386,431)(29,48,226,72,91,165,301,210,387,432)(30,37,227,61,92,166,302,211,388,421)(31,38,228,62,93,167,303,212,389,422)(32,39,217,63,94,168,304,213,390,423)(33,40,218,64,95,157,305,214,391,424)(34,41,219,65,96,158,306,215,392,425)(35,42,220,66,85,159,307,216,393,426)(36,43,221,67,86,160,308,205,394,427)(73,243,235,114,350,470,337,182,367,156)(74,244,236,115,351,471,338,183,368,145)(75,245,237,116,352,472,339,184,369,146)(76,246,238,117,353,473,340,185,370,147)(77,247,239,118,354,474,341,186,371,148)(78,248,240,119,355,475,342,187,372,149)(79,249,229,120,356,476,343,188,361,150)(80,250,230,109,357,477,344,189,362,151)(81,251,231,110,358,478,345,190,363,152)(82,252,232,111,359,479,346,191,364,153)(83,241,233,112,360,480,347,192,365,154)(84,242,234,113,349,469,348,181,366,155), (1,103,405,459)(2,460,406,104)(3,105,407,461)(4,462,408,106)(5,107,397,463)(6,464,398,108)(7,97,399,465)(8,466,400,98)(9,99,401,467)(10,468,402,100)(11,101,403,457)(12,458,404,102)(13,382,320,202)(14,203,321,383)(15,384,322,204)(16,193,323,373)(17,374,324,194)(18,195,313,375)(19,376,314,196)(20,197,315,377)(21,378,316,198)(22,199,317,379)(23,380,318,200)(24,201,319,381)(25,351,161,145)(26,146,162,352)(27,353,163,147)(28,148,164,354)(29,355,165,149)(30,150,166,356)(31,357,167,151)(32,152,168,358)(33,359,157,153)(34,154,158,360)(35,349,159,155)(36,156,160,350)(37,361,302,120)(38,109,303,362)(39,363,304,110)(40,111,305,364)(41,365,306,112)(42,113,307,366)(43,367,308,114)(44,115,309,368)(45,369,310,116)(46,117,311,370)(47,371,312,118)(48,119,301,372)(49,420,288,268)(50,269,277,409)(51,410,278,270)(52,271,279,411)(53,412,280,272)(54,273,281,413)(55,414,282,274)(56,275,283,415)(57,416,284,276)(58,265,285,417)(59,418,286,266)(60,267,287,419)(61,343,388,249)(62,250,389,344)(63,345,390,251)(64,252,391,346)(65,347,392,241)(66,242,393,348)(67,337,394,243)(68,244,395,338)(69,339,396,245)(70,246,385,340)(71,341,386,247)(72,248,387,342)(73,86,470,427)(74,428,471,87)(75,88,472,429)(76,430,473,89)(77,90,474,431)(78,432,475,91)(79,92,476,421)(80,422,477,93)(81,94,478,423)(82,424,479,95)(83,96,480,425)(84,426,469,85)(121,141,444,336)(122,325,433,142)(123,143,434,326)(124,327,435,144)(125,133,436,328)(126,329,437,134)(127,135,438,330)(128,331,439,136)(129,137,440,332)(130,333,441,138)(131,139,442,334)(132,335,443,140)(169,451,300,255)(170,256,289,452)(171,453,290,257)(172,258,291,454)(173,455,292,259)(174,260,293,456)(175,445,294,261)(176,262,295,446)(177,447,296,263)(178,264,297,448)(179,449,298,253)(180,254,299,450)(181,220,234,216)(182,205,235,221)(183,222,236,206)(184,207,237,223)(185,224,238,208)(186,209,239,225)(187,226,240,210)(188,211,229,227)(189,228,230,212)(190,213,231,217)(191,218,232,214)(192,215,233,219), (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), (1,89,399,424)(2,88,400,423)(3,87,401,422)(4,86,402,421)(5,85,403,432)(6,96,404,431)(7,95,405,430)(8,94,406,429)(9,93,407,428)(10,92,408,427)(11,91,397,426)(12,90,398,425)(13,352,314,152)(14,351,315,151)(15,350,316,150)(16,349,317,149)(17,360,318,148)(18,359,319,147)(19,358,320,146)(20,357,321,145)(21,356,322,156)(22,355,323,155)(23,354,324,154)(24,353,313,153)(25,197,167,383)(26,196,168,382)(27,195,157,381)(28,194,158,380)(29,193,159,379)(30,204,160,378)(31,203,161,377)(32,202,162,376)(33,201,163,375)(34,200,164,374)(35,199,165,373)(36,198,166,384)(37,284,308,51)(38,283,309,50)(39,282,310,49)(40,281,311,60)(41,280,312,59)(42,279,301,58)(43,278,302,57)(44,277,303,56)(45,288,304,55)(46,287,305,54)(47,286,306,53)(48,285,307,52)(61,330,394,141)(62,329,395,140)(63,328,396,139)(64,327,385,138)(65,326,386,137)(66,325,387,136)(67,336,388,135)(68,335,389,134)(69,334,390,133)(70,333,391,144)(71,332,392,143)(72,331,393,142)(73,468,476,106)(74,467,477,105)(75,466,478,104)(76,465,479,103)(77,464,480,102)(78,463,469,101)(79,462,470,100)(80,461,471,99)(81,460,472,98)(82,459,473,97)(83,458,474,108)(84,457,475,107)(109,415,368,269)(110,414,369,268)(111,413,370,267)(112,412,371,266)(113,411,372,265)(114,410,361,276)(115,409,362,275)(116,420,363,274)(117,419,364,273)(118,418,365,272)(119,417,366,271)(120,416,367,270)(121,249,438,337)(122,248,439,348)(123,247,440,347)(124,246,441,346)(125,245,442,345)(126,244,443,344)(127,243,444,343)(128,242,433,342)(129,241,434,341)(130,252,435,340)(131,251,436,339)(132,250,437,338)(169,222,294,212)(170,221,295,211)(171,220,296,210)(172,219,297,209)(173,218,298,208)(174,217,299,207)(175,228,300,206)(176,227,289,205)(177,226,290,216)(178,225,291,215)(179,224,292,214)(180,223,293,213)(181,447,240,257)(182,446,229,256)(183,445,230,255)(184,456,231,254)(185,455,232,253)(186,454,233,264)(187,453,234,263)(188,452,235,262)(189,451,236,261)(190,450,237,260)(191,449,238,259)(192,448,239,258) );
G=PermutationGroup([[(1,201,281,298,327,405,381,54,179,144),(2,202,282,299,328,406,382,55,180,133),(3,203,283,300,329,407,383,56,169,134),(4,204,284,289,330,408,384,57,170,135),(5,193,285,290,331,397,373,58,171,136),(6,194,286,291,332,398,374,59,172,137),(7,195,287,292,333,399,375,60,173,138),(8,196,288,293,334,400,376,49,174,139),(9,197,277,294,335,401,377,50,175,140),(10,198,278,295,336,402,378,51,176,141),(11,199,279,296,325,403,379,52,177,142),(12,200,280,297,326,404,380,53,178,143),(13,460,436,254,414,320,104,125,450,274),(14,461,437,255,415,321,105,126,451,275),(15,462,438,256,416,322,106,127,452,276),(16,463,439,257,417,323,107,128,453,265),(17,464,440,258,418,324,108,129,454,266),(18,465,441,259,419,313,97,130,455,267),(19,466,442,260,420,314,98,131,456,268),(20,467,443,261,409,315,99,132,445,269),(21,468,444,262,410,316,100,121,446,270),(22,457,433,263,411,317,101,122,447,271),(23,458,434,264,412,318,102,123,448,272),(24,459,435,253,413,319,103,124,449,273),(25,44,222,68,87,161,309,206,395,428),(26,45,223,69,88,162,310,207,396,429),(27,46,224,70,89,163,311,208,385,430),(28,47,225,71,90,164,312,209,386,431),(29,48,226,72,91,165,301,210,387,432),(30,37,227,61,92,166,302,211,388,421),(31,38,228,62,93,167,303,212,389,422),(32,39,217,63,94,168,304,213,390,423),(33,40,218,64,95,157,305,214,391,424),(34,41,219,65,96,158,306,215,392,425),(35,42,220,66,85,159,307,216,393,426),(36,43,221,67,86,160,308,205,394,427),(73,243,235,114,350,470,337,182,367,156),(74,244,236,115,351,471,338,183,368,145),(75,245,237,116,352,472,339,184,369,146),(76,246,238,117,353,473,340,185,370,147),(77,247,239,118,354,474,341,186,371,148),(78,248,240,119,355,475,342,187,372,149),(79,249,229,120,356,476,343,188,361,150),(80,250,230,109,357,477,344,189,362,151),(81,251,231,110,358,478,345,190,363,152),(82,252,232,111,359,479,346,191,364,153),(83,241,233,112,360,480,347,192,365,154),(84,242,234,113,349,469,348,181,366,155)], [(1,103,405,459),(2,460,406,104),(3,105,407,461),(4,462,408,106),(5,107,397,463),(6,464,398,108),(7,97,399,465),(8,466,400,98),(9,99,401,467),(10,468,402,100),(11,101,403,457),(12,458,404,102),(13,382,320,202),(14,203,321,383),(15,384,322,204),(16,193,323,373),(17,374,324,194),(18,195,313,375),(19,376,314,196),(20,197,315,377),(21,378,316,198),(22,199,317,379),(23,380,318,200),(24,201,319,381),(25,351,161,145),(26,146,162,352),(27,353,163,147),(28,148,164,354),(29,355,165,149),(30,150,166,356),(31,357,167,151),(32,152,168,358),(33,359,157,153),(34,154,158,360),(35,349,159,155),(36,156,160,350),(37,361,302,120),(38,109,303,362),(39,363,304,110),(40,111,305,364),(41,365,306,112),(42,113,307,366),(43,367,308,114),(44,115,309,368),(45,369,310,116),(46,117,311,370),(47,371,312,118),(48,119,301,372),(49,420,288,268),(50,269,277,409),(51,410,278,270),(52,271,279,411),(53,412,280,272),(54,273,281,413),(55,414,282,274),(56,275,283,415),(57,416,284,276),(58,265,285,417),(59,418,286,266),(60,267,287,419),(61,343,388,249),(62,250,389,344),(63,345,390,251),(64,252,391,346),(65,347,392,241),(66,242,393,348),(67,337,394,243),(68,244,395,338),(69,339,396,245),(70,246,385,340),(71,341,386,247),(72,248,387,342),(73,86,470,427),(74,428,471,87),(75,88,472,429),(76,430,473,89),(77,90,474,431),(78,432,475,91),(79,92,476,421),(80,422,477,93),(81,94,478,423),(82,424,479,95),(83,96,480,425),(84,426,469,85),(121,141,444,336),(122,325,433,142),(123,143,434,326),(124,327,435,144),(125,133,436,328),(126,329,437,134),(127,135,438,330),(128,331,439,136),(129,137,440,332),(130,333,441,138),(131,139,442,334),(132,335,443,140),(169,451,300,255),(170,256,289,452),(171,453,290,257),(172,258,291,454),(173,455,292,259),(174,260,293,456),(175,445,294,261),(176,262,295,446),(177,447,296,263),(178,264,297,448),(179,449,298,253),(180,254,299,450),(181,220,234,216),(182,205,235,221),(183,222,236,206),(184,207,237,223),(185,224,238,208),(186,209,239,225),(187,226,240,210),(188,211,229,227),(189,228,230,212),(190,213,231,217),(191,218,232,214),(192,215,233,219)], [(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)], [(1,89,399,424),(2,88,400,423),(3,87,401,422),(4,86,402,421),(5,85,403,432),(6,96,404,431),(7,95,405,430),(8,94,406,429),(9,93,407,428),(10,92,408,427),(11,91,397,426),(12,90,398,425),(13,352,314,152),(14,351,315,151),(15,350,316,150),(16,349,317,149),(17,360,318,148),(18,359,319,147),(19,358,320,146),(20,357,321,145),(21,356,322,156),(22,355,323,155),(23,354,324,154),(24,353,313,153),(25,197,167,383),(26,196,168,382),(27,195,157,381),(28,194,158,380),(29,193,159,379),(30,204,160,378),(31,203,161,377),(32,202,162,376),(33,201,163,375),(34,200,164,374),(35,199,165,373),(36,198,166,384),(37,284,308,51),(38,283,309,50),(39,282,310,49),(40,281,311,60),(41,280,312,59),(42,279,301,58),(43,278,302,57),(44,277,303,56),(45,288,304,55),(46,287,305,54),(47,286,306,53),(48,285,307,52),(61,330,394,141),(62,329,395,140),(63,328,396,139),(64,327,385,138),(65,326,386,137),(66,325,387,136),(67,336,388,135),(68,335,389,134),(69,334,390,133),(70,333,391,144),(71,332,392,143),(72,331,393,142),(73,468,476,106),(74,467,477,105),(75,466,478,104),(76,465,479,103),(77,464,480,102),(78,463,469,101),(79,462,470,100),(80,461,471,99),(81,460,472,98),(82,459,473,97),(83,458,474,108),(84,457,475,107),(109,415,368,269),(110,414,369,268),(111,413,370,267),(112,412,371,266),(113,411,372,265),(114,410,361,276),(115,409,362,275),(116,420,363,274),(117,419,364,273),(118,418,365,272),(119,417,366,271),(120,416,367,270),(121,249,438,337),(122,248,439,348),(123,247,440,347),(124,246,441,346),(125,245,442,345),(126,244,443,344),(127,243,444,343),(128,242,433,342),(129,241,434,341),(130,252,435,340),(131,251,436,339),(132,250,437,338),(169,222,294,212),(170,221,295,211),(171,220,296,210),(172,219,297,209),(173,218,298,208),(174,217,299,207),(175,228,300,206),(176,227,289,205),(177,226,290,216),(178,225,291,215),(179,224,292,214),(180,223,293,213),(181,447,240,257),(182,446,229,256),(183,445,230,255),(184,456,231,254),(185,455,232,253),(186,454,233,264),(187,453,234,263),(188,452,235,262),(189,451,236,261),(190,450,237,260),(191,449,238,259),(192,448,239,258)]])
60 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 | 12F | 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 | 4 | 6 | 6 | 10 | 10 | 12 | 20 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | Dic6 | C4○D20 | D4⋊2S3 | Q8⋊3S3 | S3×D5 | D4⋊2D5 | Q8×D5 | C2×S3×D5 | D5×Dic6 | D12⋊D5 | Dic5.D6 |
| kernel | Dic5.2Dic6 | Dic3×Dic5 | C30.Q8 | C6.Dic10 | C3×C10.D4 | C5×C4⋊Dic3 | C30.4Q8 | C10.D4 | C3×Dic5 | C4⋊Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | Dic5 | C6 | C10 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of Dic5.2Dic6 ►in GL6(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 27 | 25 | 0 | 0 | 0 | 0 |
| 27 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 47 | 0 | 0 |
| 0 | 0 | 39 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 59 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 53 | 7 |
| 0 | 0 | 0 | 0 | 26 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 0 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 11 |
| 0 | 0 | 0 | 0 | 21 | 45 |
G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,43,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,27,0,0,0,0,25,34,0,0,0,0,0,0,39,39,0,0,0,0,47,22,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,59,60,0,0,0,0,0,0,53,26,0,0,0,0,7,0],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,16,21,0,0,0,0,11,45] >;
Dic5.2Dic6 in GAP, Magma, Sage, TeX
{\rm Dic}_5._2{\rm Dic}_6 % in TeX
G:=Group("Dic5.2Dic6"); // GroupNames label
G:=SmallGroup(480,411);
// by ID
G=gap.SmallGroup(480,411);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,422,303,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^10=c^12=1,b^2=a^5,d^2=a^5*c^6,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations