direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12⋊2Q8, C60⋊11Q8, C20⋊9Dic6, C20.42D12, C60.180D4, C12⋊2(C5×Q8), C15⋊12(C4⋊Q8), C4⋊2(C5×Dic6), C6.1(D4×C10), C4.4(C5×D12), C6.2(Q8×C10), (C4×C12).2C10, (C4×C20).10S3, (C4×C60).12C2, C12.27(C5×D4), C2.4(C10×D12), C42.4(C5×S3), C30.83(C2×Q8), C10.72(C2×D12), (C2×C20).426D6, C30.288(C2×D4), C4⋊Dic3.4C10, C2.4(C10×Dic6), (C2×Dic6).2C10, C10.42(C2×Dic6), (C2×C60).525C22, (C2×C30).389C23, (C10×Dic6).12C2, (C10×Dic3).134C22, C3⋊1(C5×C4⋊Q8), (C2×C4).76(S3×C10), C22.34(S3×C2×C10), (C2×C12).89(C2×C10), (C5×C4⋊Dic3).18C2, (C2×C6).10(C22×C10), (C2×Dic3).1(C2×C10), (C2×C10).323(C22×S3), SmallGroup(480,748)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C12⋊2Q8
G = < a,b,c,d | a5=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 276 in 136 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, C20, C20, C2×C10, Dic6, C2×Dic3, C2×C12, C2×C12, C30, C30, C4⋊Q8, C2×C20, C2×C20, C2×C20, C5×Q8, C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3, C60, C2×C30, C4×C20, C5×C4⋊C4, Q8×C10, C12⋊2Q8, C5×Dic6, C10×Dic3, C2×C60, C2×C60, C5×C4⋊Q8, C5×C4⋊Dic3, C4×C60, C10×Dic6, C5×C12⋊2Q8
Quotients: C1, C2, C22, C5, S3, D4, Q8, C23, C10, D6, C2×D4, C2×Q8, C2×C10, Dic6, D12, C22×S3, C5×S3, C4⋊Q8, C5×D4, C5×Q8, C22×C10, C2×Dic6, C2×D12, S3×C10, D4×C10, Q8×C10, C12⋊2Q8, C5×Dic6, C5×D12, S3×C2×C10, C5×C4⋊Q8, C10×Dic6, C10×D12, C5×C12⋊2Q8
►Regular action on 480 points
Generators in S480
(1 66 186 139 146)(2 67 187 140 147)(3 68 188 141 148)(4 69 189 142 149)(5 70 190 143 150)(6 71 191 144 151)(7 72 192 133 152)(8 61 181 134 153)(9 62 182 135 154)(10 63 183 136 155)(11 64 184 137 156)(12 65 185 138 145)(13 205 431 417 131)(14 206 432 418 132)(15 207 421 419 121)(16 208 422 420 122)(17 209 423 409 123)(18 210 424 410 124)(19 211 425 411 125)(20 212 426 412 126)(21 213 427 413 127)(22 214 428 414 128)(23 215 429 415 129)(24 216 430 416 130)(25 42 444 231 298)(26 43 433 232 299)(27 44 434 233 300)(28 45 435 234 289)(29 46 436 235 290)(30 47 437 236 291)(31 48 438 237 292)(32 37 439 238 293)(33 38 440 239 294)(34 39 441 240 295)(35 40 442 229 296)(36 41 443 230 297)(49 245 103 355 267)(50 246 104 356 268)(51 247 105 357 269)(52 248 106 358 270)(53 249 107 359 271)(54 250 108 360 272)(55 251 97 349 273)(56 252 98 350 274)(57 241 99 351 275)(58 242 100 352 276)(59 243 101 353 265)(60 244 102 354 266)(73 365 394 459 449)(74 366 395 460 450)(75 367 396 461 451)(76 368 385 462 452)(77 369 386 463 453)(78 370 387 464 454)(79 371 388 465 455)(80 372 389 466 456)(81 361 390 467 445)(82 362 391 468 446)(83 363 392 457 447)(84 364 393 458 448)(85 319 472 164 179)(86 320 473 165 180)(87 321 474 166 169)(88 322 475 167 170)(89 323 476 168 171)(90 324 477 157 172)(91 313 478 158 173)(92 314 479 159 174)(93 315 480 160 175)(94 316 469 161 176)(95 317 470 162 177)(96 318 471 163 178)(109 261 398 333 196)(110 262 399 334 197)(111 263 400 335 198)(112 264 401 336 199)(113 253 402 325 200)(114 254 403 326 201)(115 255 404 327 202)(116 256 405 328 203)(117 257 406 329 204)(118 258 407 330 193)(119 259 408 331 194)(120 260 397 332 195)(217 380 278 304 340)(218 381 279 305 341)(219 382 280 306 342)(220 383 281 307 343)(221 384 282 308 344)(222 373 283 309 345)(223 374 284 310 346)(224 375 285 311 347)(225 376 286 312 348)(226 377 287 301 337)(227 378 288 302 338)(228 379 277 303 339)
(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)
(1 378 469 83)(2 379 470 84)(3 380 471 73)(4 381 472 74)(5 382 473 75)(6 383 474 76)(7 384 475 77)(8 373 476 78)(9 374 477 79)(10 375 478 80)(11 376 479 81)(12 377 480 82)(13 28 406 275)(14 29 407 276)(15 30 408 265)(16 31 397 266)(17 32 398 267)(18 33 399 268)(19 34 400 269)(20 35 401 270)(21 36 402 271)(22 25 403 272)(23 26 404 273)(24 27 405 274)(37 333 49 209)(38 334 50 210)(39 335 51 211)(40 336 52 212)(41 325 53 213)(42 326 54 214)(43 327 55 215)(44 328 56 216)(45 329 57 205)(46 330 58 206)(47 331 59 207)(48 332 60 208)(61 283 168 370)(62 284 157 371)(63 285 158 372)(64 286 159 361)(65 287 160 362)(66 288 161 363)(67 277 162 364)(68 278 163 365)(69 279 164 366)(70 280 165 367)(71 281 166 368)(72 282 167 369)(85 460 142 341)(86 461 143 342)(87 462 144 343)(88 463 133 344)(89 464 134 345)(90 465 135 346)(91 466 136 347)(92 467 137 348)(93 468 138 337)(94 457 139 338)(95 458 140 339)(96 459 141 340)(97 415 232 115)(98 416 233 116)(99 417 234 117)(100 418 235 118)(101 419 236 119)(102 420 237 120)(103 409 238 109)(104 410 239 110)(105 411 240 111)(106 412 229 112)(107 413 230 113)(108 414 231 114)(121 291 259 353)(122 292 260 354)(123 293 261 355)(124 294 262 356)(125 295 263 357)(126 296 264 358)(127 297 253 359)(128 298 254 360)(129 299 255 349)(130 300 256 350)(131 289 257 351)(132 290 258 352)(145 226 315 446)(146 227 316 447)(147 228 317 448)(148 217 318 449)(149 218 319 450)(150 219 320 451)(151 220 321 452)(152 221 322 453)(153 222 323 454)(154 223 324 455)(155 224 313 456)(156 225 314 445)(169 385 191 307)(170 386 192 308)(171 387 181 309)(172 388 182 310)(173 389 183 311)(174 390 184 312)(175 391 185 301)(176 392 186 302)(177 393 187 303)(178 394 188 304)(179 395 189 305)(180 396 190 306)(193 242 432 436)(194 243 421 437)(195 244 422 438)(196 245 423 439)(197 246 424 440)(198 247 425 441)(199 248 426 442)(200 249 427 443)(201 250 428 444)(202 251 429 433)(203 252 430 434)(204 241 431 435)
(1 30 469 265)(2 29 470 276)(3 28 471 275)(4 27 472 274)(5 26 473 273)(6 25 474 272)(7 36 475 271)(8 35 476 270)(9 34 477 269)(10 33 478 268)(11 32 479 267)(12 31 480 266)(13 73 406 380)(14 84 407 379)(15 83 408 378)(16 82 397 377)(17 81 398 376)(18 80 399 375)(19 79 400 374)(20 78 401 373)(21 77 402 384)(22 76 403 383)(23 75 404 382)(24 74 405 381)(37 159 49 64)(38 158 50 63)(39 157 51 62)(40 168 52 61)(41 167 53 72)(42 166 54 71)(43 165 55 70)(44 164 56 69)(45 163 57 68)(46 162 58 67)(47 161 59 66)(48 160 60 65)(85 98 142 233)(86 97 143 232)(87 108 144 231)(88 107 133 230)(89 106 134 229)(90 105 135 240)(91 104 136 239)(92 103 137 238)(93 102 138 237)(94 101 139 236)(95 100 140 235)(96 99 141 234)(109 348 409 467)(110 347 410 466)(111 346 411 465)(112 345 412 464)(113 344 413 463)(114 343 414 462)(115 342 415 461)(116 341 416 460)(117 340 417 459)(118 339 418 458)(119 338 419 457)(120 337 420 468)(121 447 259 227)(122 446 260 226)(123 445 261 225)(124 456 262 224)(125 455 263 223)(126 454 264 222)(127 453 253 221)(128 452 254 220)(129 451 255 219)(130 450 256 218)(131 449 257 217)(132 448 258 228)(145 292 315 354)(146 291 316 353)(147 290 317 352)(148 289 318 351)(149 300 319 350)(150 299 320 349)(151 298 321 360)(152 297 322 359)(153 296 323 358)(154 295 324 357)(155 294 313 356)(156 293 314 355)(169 250 191 444)(170 249 192 443)(171 248 181 442)(172 247 182 441)(173 246 183 440)(174 245 184 439)(175 244 185 438)(176 243 186 437)(177 242 187 436)(178 241 188 435)(179 252 189 434)(180 251 190 433)(193 303 432 393)(194 302 421 392)(195 301 422 391)(196 312 423 390)(197 311 424 389)(198 310 425 388)(199 309 426 387)(200 308 427 386)(201 307 428 385)(202 306 429 396)(203 305 430 395)(204 304 431 394)(205 365 329 278)(206 364 330 277)(207 363 331 288)(208 362 332 287)(209 361 333 286)(210 372 334 285)(211 371 335 284)(212 370 336 283)(213 369 325 282)(214 368 326 281)(215 367 327 280)(216 366 328 279)
G:=sub<Sym(480)| (1,66,186,139,146)(2,67,187,140,147)(3,68,188,141,148)(4,69,189,142,149)(5,70,190,143,150)(6,71,191,144,151)(7,72,192,133,152)(8,61,181,134,153)(9,62,182,135,154)(10,63,183,136,155)(11,64,184,137,156)(12,65,185,138,145)(13,205,431,417,131)(14,206,432,418,132)(15,207,421,419,121)(16,208,422,420,122)(17,209,423,409,123)(18,210,424,410,124)(19,211,425,411,125)(20,212,426,412,126)(21,213,427,413,127)(22,214,428,414,128)(23,215,429,415,129)(24,216,430,416,130)(25,42,444,231,298)(26,43,433,232,299)(27,44,434,233,300)(28,45,435,234,289)(29,46,436,235,290)(30,47,437,236,291)(31,48,438,237,292)(32,37,439,238,293)(33,38,440,239,294)(34,39,441,240,295)(35,40,442,229,296)(36,41,443,230,297)(49,245,103,355,267)(50,246,104,356,268)(51,247,105,357,269)(52,248,106,358,270)(53,249,107,359,271)(54,250,108,360,272)(55,251,97,349,273)(56,252,98,350,274)(57,241,99,351,275)(58,242,100,352,276)(59,243,101,353,265)(60,244,102,354,266)(73,365,394,459,449)(74,366,395,460,450)(75,367,396,461,451)(76,368,385,462,452)(77,369,386,463,453)(78,370,387,464,454)(79,371,388,465,455)(80,372,389,466,456)(81,361,390,467,445)(82,362,391,468,446)(83,363,392,457,447)(84,364,393,458,448)(85,319,472,164,179)(86,320,473,165,180)(87,321,474,166,169)(88,322,475,167,170)(89,323,476,168,171)(90,324,477,157,172)(91,313,478,158,173)(92,314,479,159,174)(93,315,480,160,175)(94,316,469,161,176)(95,317,470,162,177)(96,318,471,163,178)(109,261,398,333,196)(110,262,399,334,197)(111,263,400,335,198)(112,264,401,336,199)(113,253,402,325,200)(114,254,403,326,201)(115,255,404,327,202)(116,256,405,328,203)(117,257,406,329,204)(118,258,407,330,193)(119,259,408,331,194)(120,260,397,332,195)(217,380,278,304,340)(218,381,279,305,341)(219,382,280,306,342)(220,383,281,307,343)(221,384,282,308,344)(222,373,283,309,345)(223,374,284,310,346)(224,375,285,311,347)(225,376,286,312,348)(226,377,287,301,337)(227,378,288,302,338)(228,379,277,303,339), (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), (1,378,469,83)(2,379,470,84)(3,380,471,73)(4,381,472,74)(5,382,473,75)(6,383,474,76)(7,384,475,77)(8,373,476,78)(9,374,477,79)(10,375,478,80)(11,376,479,81)(12,377,480,82)(13,28,406,275)(14,29,407,276)(15,30,408,265)(16,31,397,266)(17,32,398,267)(18,33,399,268)(19,34,400,269)(20,35,401,270)(21,36,402,271)(22,25,403,272)(23,26,404,273)(24,27,405,274)(37,333,49,209)(38,334,50,210)(39,335,51,211)(40,336,52,212)(41,325,53,213)(42,326,54,214)(43,327,55,215)(44,328,56,216)(45,329,57,205)(46,330,58,206)(47,331,59,207)(48,332,60,208)(61,283,168,370)(62,284,157,371)(63,285,158,372)(64,286,159,361)(65,287,160,362)(66,288,161,363)(67,277,162,364)(68,278,163,365)(69,279,164,366)(70,280,165,367)(71,281,166,368)(72,282,167,369)(85,460,142,341)(86,461,143,342)(87,462,144,343)(88,463,133,344)(89,464,134,345)(90,465,135,346)(91,466,136,347)(92,467,137,348)(93,468,138,337)(94,457,139,338)(95,458,140,339)(96,459,141,340)(97,415,232,115)(98,416,233,116)(99,417,234,117)(100,418,235,118)(101,419,236,119)(102,420,237,120)(103,409,238,109)(104,410,239,110)(105,411,240,111)(106,412,229,112)(107,413,230,113)(108,414,231,114)(121,291,259,353)(122,292,260,354)(123,293,261,355)(124,294,262,356)(125,295,263,357)(126,296,264,358)(127,297,253,359)(128,298,254,360)(129,299,255,349)(130,300,256,350)(131,289,257,351)(132,290,258,352)(145,226,315,446)(146,227,316,447)(147,228,317,448)(148,217,318,449)(149,218,319,450)(150,219,320,451)(151,220,321,452)(152,221,322,453)(153,222,323,454)(154,223,324,455)(155,224,313,456)(156,225,314,445)(169,385,191,307)(170,386,192,308)(171,387,181,309)(172,388,182,310)(173,389,183,311)(174,390,184,312)(175,391,185,301)(176,392,186,302)(177,393,187,303)(178,394,188,304)(179,395,189,305)(180,396,190,306)(193,242,432,436)(194,243,421,437)(195,244,422,438)(196,245,423,439)(197,246,424,440)(198,247,425,441)(199,248,426,442)(200,249,427,443)(201,250,428,444)(202,251,429,433)(203,252,430,434)(204,241,431,435), (1,30,469,265)(2,29,470,276)(3,28,471,275)(4,27,472,274)(5,26,473,273)(6,25,474,272)(7,36,475,271)(8,35,476,270)(9,34,477,269)(10,33,478,268)(11,32,479,267)(12,31,480,266)(13,73,406,380)(14,84,407,379)(15,83,408,378)(16,82,397,377)(17,81,398,376)(18,80,399,375)(19,79,400,374)(20,78,401,373)(21,77,402,384)(22,76,403,383)(23,75,404,382)(24,74,405,381)(37,159,49,64)(38,158,50,63)(39,157,51,62)(40,168,52,61)(41,167,53,72)(42,166,54,71)(43,165,55,70)(44,164,56,69)(45,163,57,68)(46,162,58,67)(47,161,59,66)(48,160,60,65)(85,98,142,233)(86,97,143,232)(87,108,144,231)(88,107,133,230)(89,106,134,229)(90,105,135,240)(91,104,136,239)(92,103,137,238)(93,102,138,237)(94,101,139,236)(95,100,140,235)(96,99,141,234)(109,348,409,467)(110,347,410,466)(111,346,411,465)(112,345,412,464)(113,344,413,463)(114,343,414,462)(115,342,415,461)(116,341,416,460)(117,340,417,459)(118,339,418,458)(119,338,419,457)(120,337,420,468)(121,447,259,227)(122,446,260,226)(123,445,261,225)(124,456,262,224)(125,455,263,223)(126,454,264,222)(127,453,253,221)(128,452,254,220)(129,451,255,219)(130,450,256,218)(131,449,257,217)(132,448,258,228)(145,292,315,354)(146,291,316,353)(147,290,317,352)(148,289,318,351)(149,300,319,350)(150,299,320,349)(151,298,321,360)(152,297,322,359)(153,296,323,358)(154,295,324,357)(155,294,313,356)(156,293,314,355)(169,250,191,444)(170,249,192,443)(171,248,181,442)(172,247,182,441)(173,246,183,440)(174,245,184,439)(175,244,185,438)(176,243,186,437)(177,242,187,436)(178,241,188,435)(179,252,189,434)(180,251,190,433)(193,303,432,393)(194,302,421,392)(195,301,422,391)(196,312,423,390)(197,311,424,389)(198,310,425,388)(199,309,426,387)(200,308,427,386)(201,307,428,385)(202,306,429,396)(203,305,430,395)(204,304,431,394)(205,365,329,278)(206,364,330,277)(207,363,331,288)(208,362,332,287)(209,361,333,286)(210,372,334,285)(211,371,335,284)(212,370,336,283)(213,369,325,282)(214,368,326,281)(215,367,327,280)(216,366,328,279)>;
G:=Group( (1,66,186,139,146)(2,67,187,140,147)(3,68,188,141,148)(4,69,189,142,149)(5,70,190,143,150)(6,71,191,144,151)(7,72,192,133,152)(8,61,181,134,153)(9,62,182,135,154)(10,63,183,136,155)(11,64,184,137,156)(12,65,185,138,145)(13,205,431,417,131)(14,206,432,418,132)(15,207,421,419,121)(16,208,422,420,122)(17,209,423,409,123)(18,210,424,410,124)(19,211,425,411,125)(20,212,426,412,126)(21,213,427,413,127)(22,214,428,414,128)(23,215,429,415,129)(24,216,430,416,130)(25,42,444,231,298)(26,43,433,232,299)(27,44,434,233,300)(28,45,435,234,289)(29,46,436,235,290)(30,47,437,236,291)(31,48,438,237,292)(32,37,439,238,293)(33,38,440,239,294)(34,39,441,240,295)(35,40,442,229,296)(36,41,443,230,297)(49,245,103,355,267)(50,246,104,356,268)(51,247,105,357,269)(52,248,106,358,270)(53,249,107,359,271)(54,250,108,360,272)(55,251,97,349,273)(56,252,98,350,274)(57,241,99,351,275)(58,242,100,352,276)(59,243,101,353,265)(60,244,102,354,266)(73,365,394,459,449)(74,366,395,460,450)(75,367,396,461,451)(76,368,385,462,452)(77,369,386,463,453)(78,370,387,464,454)(79,371,388,465,455)(80,372,389,466,456)(81,361,390,467,445)(82,362,391,468,446)(83,363,392,457,447)(84,364,393,458,448)(85,319,472,164,179)(86,320,473,165,180)(87,321,474,166,169)(88,322,475,167,170)(89,323,476,168,171)(90,324,477,157,172)(91,313,478,158,173)(92,314,479,159,174)(93,315,480,160,175)(94,316,469,161,176)(95,317,470,162,177)(96,318,471,163,178)(109,261,398,333,196)(110,262,399,334,197)(111,263,400,335,198)(112,264,401,336,199)(113,253,402,325,200)(114,254,403,326,201)(115,255,404,327,202)(116,256,405,328,203)(117,257,406,329,204)(118,258,407,330,193)(119,259,408,331,194)(120,260,397,332,195)(217,380,278,304,340)(218,381,279,305,341)(219,382,280,306,342)(220,383,281,307,343)(221,384,282,308,344)(222,373,283,309,345)(223,374,284,310,346)(224,375,285,311,347)(225,376,286,312,348)(226,377,287,301,337)(227,378,288,302,338)(228,379,277,303,339), (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), (1,378,469,83)(2,379,470,84)(3,380,471,73)(4,381,472,74)(5,382,473,75)(6,383,474,76)(7,384,475,77)(8,373,476,78)(9,374,477,79)(10,375,478,80)(11,376,479,81)(12,377,480,82)(13,28,406,275)(14,29,407,276)(15,30,408,265)(16,31,397,266)(17,32,398,267)(18,33,399,268)(19,34,400,269)(20,35,401,270)(21,36,402,271)(22,25,403,272)(23,26,404,273)(24,27,405,274)(37,333,49,209)(38,334,50,210)(39,335,51,211)(40,336,52,212)(41,325,53,213)(42,326,54,214)(43,327,55,215)(44,328,56,216)(45,329,57,205)(46,330,58,206)(47,331,59,207)(48,332,60,208)(61,283,168,370)(62,284,157,371)(63,285,158,372)(64,286,159,361)(65,287,160,362)(66,288,161,363)(67,277,162,364)(68,278,163,365)(69,279,164,366)(70,280,165,367)(71,281,166,368)(72,282,167,369)(85,460,142,341)(86,461,143,342)(87,462,144,343)(88,463,133,344)(89,464,134,345)(90,465,135,346)(91,466,136,347)(92,467,137,348)(93,468,138,337)(94,457,139,338)(95,458,140,339)(96,459,141,340)(97,415,232,115)(98,416,233,116)(99,417,234,117)(100,418,235,118)(101,419,236,119)(102,420,237,120)(103,409,238,109)(104,410,239,110)(105,411,240,111)(106,412,229,112)(107,413,230,113)(108,414,231,114)(121,291,259,353)(122,292,260,354)(123,293,261,355)(124,294,262,356)(125,295,263,357)(126,296,264,358)(127,297,253,359)(128,298,254,360)(129,299,255,349)(130,300,256,350)(131,289,257,351)(132,290,258,352)(145,226,315,446)(146,227,316,447)(147,228,317,448)(148,217,318,449)(149,218,319,450)(150,219,320,451)(151,220,321,452)(152,221,322,453)(153,222,323,454)(154,223,324,455)(155,224,313,456)(156,225,314,445)(169,385,191,307)(170,386,192,308)(171,387,181,309)(172,388,182,310)(173,389,183,311)(174,390,184,312)(175,391,185,301)(176,392,186,302)(177,393,187,303)(178,394,188,304)(179,395,189,305)(180,396,190,306)(193,242,432,436)(194,243,421,437)(195,244,422,438)(196,245,423,439)(197,246,424,440)(198,247,425,441)(199,248,426,442)(200,249,427,443)(201,250,428,444)(202,251,429,433)(203,252,430,434)(204,241,431,435), (1,30,469,265)(2,29,470,276)(3,28,471,275)(4,27,472,274)(5,26,473,273)(6,25,474,272)(7,36,475,271)(8,35,476,270)(9,34,477,269)(10,33,478,268)(11,32,479,267)(12,31,480,266)(13,73,406,380)(14,84,407,379)(15,83,408,378)(16,82,397,377)(17,81,398,376)(18,80,399,375)(19,79,400,374)(20,78,401,373)(21,77,402,384)(22,76,403,383)(23,75,404,382)(24,74,405,381)(37,159,49,64)(38,158,50,63)(39,157,51,62)(40,168,52,61)(41,167,53,72)(42,166,54,71)(43,165,55,70)(44,164,56,69)(45,163,57,68)(46,162,58,67)(47,161,59,66)(48,160,60,65)(85,98,142,233)(86,97,143,232)(87,108,144,231)(88,107,133,230)(89,106,134,229)(90,105,135,240)(91,104,136,239)(92,103,137,238)(93,102,138,237)(94,101,139,236)(95,100,140,235)(96,99,141,234)(109,348,409,467)(110,347,410,466)(111,346,411,465)(112,345,412,464)(113,344,413,463)(114,343,414,462)(115,342,415,461)(116,341,416,460)(117,340,417,459)(118,339,418,458)(119,338,419,457)(120,337,420,468)(121,447,259,227)(122,446,260,226)(123,445,261,225)(124,456,262,224)(125,455,263,223)(126,454,264,222)(127,453,253,221)(128,452,254,220)(129,451,255,219)(130,450,256,218)(131,449,257,217)(132,448,258,228)(145,292,315,354)(146,291,316,353)(147,290,317,352)(148,289,318,351)(149,300,319,350)(150,299,320,349)(151,298,321,360)(152,297,322,359)(153,296,323,358)(154,295,324,357)(155,294,313,356)(156,293,314,355)(169,250,191,444)(170,249,192,443)(171,248,181,442)(172,247,182,441)(173,246,183,440)(174,245,184,439)(175,244,185,438)(176,243,186,437)(177,242,187,436)(178,241,188,435)(179,252,189,434)(180,251,190,433)(193,303,432,393)(194,302,421,392)(195,301,422,391)(196,312,423,390)(197,311,424,389)(198,310,425,388)(199,309,426,387)(200,308,427,386)(201,307,428,385)(202,306,429,396)(203,305,430,395)(204,304,431,394)(205,365,329,278)(206,364,330,277)(207,363,331,288)(208,362,332,287)(209,361,333,286)(210,372,334,285)(211,371,335,284)(212,370,336,283)(213,369,325,282)(214,368,326,281)(215,367,327,280)(216,366,328,279) );
G=PermutationGroup([[(1,66,186,139,146),(2,67,187,140,147),(3,68,188,141,148),(4,69,189,142,149),(5,70,190,143,150),(6,71,191,144,151),(7,72,192,133,152),(8,61,181,134,153),(9,62,182,135,154),(10,63,183,136,155),(11,64,184,137,156),(12,65,185,138,145),(13,205,431,417,131),(14,206,432,418,132),(15,207,421,419,121),(16,208,422,420,122),(17,209,423,409,123),(18,210,424,410,124),(19,211,425,411,125),(20,212,426,412,126),(21,213,427,413,127),(22,214,428,414,128),(23,215,429,415,129),(24,216,430,416,130),(25,42,444,231,298),(26,43,433,232,299),(27,44,434,233,300),(28,45,435,234,289),(29,46,436,235,290),(30,47,437,236,291),(31,48,438,237,292),(32,37,439,238,293),(33,38,440,239,294),(34,39,441,240,295),(35,40,442,229,296),(36,41,443,230,297),(49,245,103,355,267),(50,246,104,356,268),(51,247,105,357,269),(52,248,106,358,270),(53,249,107,359,271),(54,250,108,360,272),(55,251,97,349,273),(56,252,98,350,274),(57,241,99,351,275),(58,242,100,352,276),(59,243,101,353,265),(60,244,102,354,266),(73,365,394,459,449),(74,366,395,460,450),(75,367,396,461,451),(76,368,385,462,452),(77,369,386,463,453),(78,370,387,464,454),(79,371,388,465,455),(80,372,389,466,456),(81,361,390,467,445),(82,362,391,468,446),(83,363,392,457,447),(84,364,393,458,448),(85,319,472,164,179),(86,320,473,165,180),(87,321,474,166,169),(88,322,475,167,170),(89,323,476,168,171),(90,324,477,157,172),(91,313,478,158,173),(92,314,479,159,174),(93,315,480,160,175),(94,316,469,161,176),(95,317,470,162,177),(96,318,471,163,178),(109,261,398,333,196),(110,262,399,334,197),(111,263,400,335,198),(112,264,401,336,199),(113,253,402,325,200),(114,254,403,326,201),(115,255,404,327,202),(116,256,405,328,203),(117,257,406,329,204),(118,258,407,330,193),(119,259,408,331,194),(120,260,397,332,195),(217,380,278,304,340),(218,381,279,305,341),(219,382,280,306,342),(220,383,281,307,343),(221,384,282,308,344),(222,373,283,309,345),(223,374,284,310,346),(224,375,285,311,347),(225,376,286,312,348),(226,377,287,301,337),(227,378,288,302,338),(228,379,277,303,339)], [(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)], [(1,378,469,83),(2,379,470,84),(3,380,471,73),(4,381,472,74),(5,382,473,75),(6,383,474,76),(7,384,475,77),(8,373,476,78),(9,374,477,79),(10,375,478,80),(11,376,479,81),(12,377,480,82),(13,28,406,275),(14,29,407,276),(15,30,408,265),(16,31,397,266),(17,32,398,267),(18,33,399,268),(19,34,400,269),(20,35,401,270),(21,36,402,271),(22,25,403,272),(23,26,404,273),(24,27,405,274),(37,333,49,209),(38,334,50,210),(39,335,51,211),(40,336,52,212),(41,325,53,213),(42,326,54,214),(43,327,55,215),(44,328,56,216),(45,329,57,205),(46,330,58,206),(47,331,59,207),(48,332,60,208),(61,283,168,370),(62,284,157,371),(63,285,158,372),(64,286,159,361),(65,287,160,362),(66,288,161,363),(67,277,162,364),(68,278,163,365),(69,279,164,366),(70,280,165,367),(71,281,166,368),(72,282,167,369),(85,460,142,341),(86,461,143,342),(87,462,144,343),(88,463,133,344),(89,464,134,345),(90,465,135,346),(91,466,136,347),(92,467,137,348),(93,468,138,337),(94,457,139,338),(95,458,140,339),(96,459,141,340),(97,415,232,115),(98,416,233,116),(99,417,234,117),(100,418,235,118),(101,419,236,119),(102,420,237,120),(103,409,238,109),(104,410,239,110),(105,411,240,111),(106,412,229,112),(107,413,230,113),(108,414,231,114),(121,291,259,353),(122,292,260,354),(123,293,261,355),(124,294,262,356),(125,295,263,357),(126,296,264,358),(127,297,253,359),(128,298,254,360),(129,299,255,349),(130,300,256,350),(131,289,257,351),(132,290,258,352),(145,226,315,446),(146,227,316,447),(147,228,317,448),(148,217,318,449),(149,218,319,450),(150,219,320,451),(151,220,321,452),(152,221,322,453),(153,222,323,454),(154,223,324,455),(155,224,313,456),(156,225,314,445),(169,385,191,307),(170,386,192,308),(171,387,181,309),(172,388,182,310),(173,389,183,311),(174,390,184,312),(175,391,185,301),(176,392,186,302),(177,393,187,303),(178,394,188,304),(179,395,189,305),(180,396,190,306),(193,242,432,436),(194,243,421,437),(195,244,422,438),(196,245,423,439),(197,246,424,440),(198,247,425,441),(199,248,426,442),(200,249,427,443),(201,250,428,444),(202,251,429,433),(203,252,430,434),(204,241,431,435)], [(1,30,469,265),(2,29,470,276),(3,28,471,275),(4,27,472,274),(5,26,473,273),(6,25,474,272),(7,36,475,271),(8,35,476,270),(9,34,477,269),(10,33,478,268),(11,32,479,267),(12,31,480,266),(13,73,406,380),(14,84,407,379),(15,83,408,378),(16,82,397,377),(17,81,398,376),(18,80,399,375),(19,79,400,374),(20,78,401,373),(21,77,402,384),(22,76,403,383),(23,75,404,382),(24,74,405,381),(37,159,49,64),(38,158,50,63),(39,157,51,62),(40,168,52,61),(41,167,53,72),(42,166,54,71),(43,165,55,70),(44,164,56,69),(45,163,57,68),(46,162,58,67),(47,161,59,66),(48,160,60,65),(85,98,142,233),(86,97,143,232),(87,108,144,231),(88,107,133,230),(89,106,134,229),(90,105,135,240),(91,104,136,239),(92,103,137,238),(93,102,138,237),(94,101,139,236),(95,100,140,235),(96,99,141,234),(109,348,409,467),(110,347,410,466),(111,346,411,465),(112,345,412,464),(113,344,413,463),(114,343,414,462),(115,342,415,461),(116,341,416,460),(117,340,417,459),(118,339,418,458),(119,338,419,457),(120,337,420,468),(121,447,259,227),(122,446,260,226),(123,445,261,225),(124,456,262,224),(125,455,263,223),(126,454,264,222),(127,453,253,221),(128,452,254,220),(129,451,255,219),(130,450,256,218),(131,449,257,217),(132,448,258,228),(145,292,315,354),(146,291,316,353),(147,290,317,352),(148,289,318,351),(149,300,319,350),(150,299,320,349),(151,298,321,360),(152,297,322,359),(153,296,323,358),(154,295,324,357),(155,294,313,356),(156,293,314,355),(169,250,191,444),(170,249,192,443),(171,248,181,442),(172,247,182,441),(173,246,183,440),(174,245,184,439),(175,244,185,438),(176,243,186,437),(177,242,187,436),(178,241,188,435),(179,252,189,434),(180,251,190,433),(193,303,432,393),(194,302,421,392),(195,301,422,391),(196,312,423,390),(197,311,424,389),(198,310,425,388),(199,309,426,387),(200,308,427,386),(201,307,428,385),(202,306,429,396),(203,305,430,395),(204,304,431,394),(205,365,329,278),(206,364,330,277),(207,363,331,288),(208,362,332,287),(209,361,333,286),(210,372,334,285),(211,371,335,284),(212,370,336,283),(213,369,325,282),(214,368,326,281),(215,367,327,280),(216,366,328,279)]])
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 20Y | ··· | 20AN | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D4 | Q8 | D6 | Dic6 | D12 | C5×S3 | C5×D4 | C5×Q8 | S3×C10 | C5×Dic6 | C5×D12 |
| kernel | C5×C12⋊2Q8 | C5×C4⋊Dic3 | C4×C60 | C10×Dic6 | C12⋊2Q8 | C4⋊Dic3 | C4×C12 | C2×Dic6 | C4×C20 | C60 | C60 | C2×C20 | C20 | C20 | C42 | C12 | C12 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 4 | 16 | 4 | 8 | 1 | 2 | 4 | 3 | 8 | 4 | 4 | 8 | 16 | 12 | 32 | 16 |
Matrix representation of C5×C12⋊2Q8 ►in GL4(𝔽61) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 23 | 38 | 0 | 0 |
| 23 | 46 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 60 | 0 |
| 23 | 46 | 0 | 0 |
| 15 | 38 | 0 | 0 |
| 0 | 0 | 23 | 46 |
| 0 | 0 | 15 | 38 |
| 43 | 52 | 0 | 0 |
| 9 | 18 | 0 | 0 |
| 0 | 0 | 8 | 49 |
| 0 | 0 | 41 | 53 |
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,58,0,0,0,0,58],[23,23,0,0,38,46,0,0,0,0,1,60,0,0,1,0],[23,15,0,0,46,38,0,0,0,0,23,15,0,0,46,38],[43,9,0,0,52,18,0,0,0,0,8,41,0,0,49,53] >;
C5×C12⋊2Q8 in GAP, Magma, Sage, TeX
C_5\times C_{12}\rtimes_2Q_8 % in TeX
G:=Group("C5xC12:2Q8"); // GroupNames label
G:=SmallGroup(480,748);
// by ID
G=gap.SmallGroup(480,748);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,1149,568,926,226,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^12=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations