direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12.3Q8, C60.21Q8, C20.18Dic6, C6.6(Q8×C10), C12.3(C5×Q8), C30.87(C2×Q8), C4.3(C5×Dic6), (C2×C20).237D6, C4⋊Dic3.7C10, C2.8(C10×Dic6), Dic3⋊C4.4C10, (C4×Dic3).3C10, C10.46(C2×Dic6), C15⋊21(C42.C2), C30.249(C4○D4), (C2×C30).410C23, (C2×C60).351C22, (Dic3×C20).12C2, C10.47(Q8⋊3S3), C10.115(D4⋊2S3), (C10×Dic3).217C22, C4⋊C4.6(C5×S3), (C3×C4⋊C4).7C10, (C5×C4⋊C4).13S3, C3⋊3(C5×C42.C2), C6.24(C5×C4○D4), (C15×C4⋊C4).21C2, (C2×C4).43(S3×C10), C22.48(S3×C2×C10), C2.4(C5×Q8⋊3S3), (C2×C12).20(C2×C10), C2.12(C5×D4⋊2S3), (C5×C4⋊Dic3).21C2, (C5×Dic3⋊C4).11C2, (C2×C6).31(C22×C10), (C2×C10).344(C22×S3), (C2×Dic3).25(C2×C10), SmallGroup(480,769)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C12.3Q8
G = < a,b,c,d | a5=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b6c-1 >
Subgroups: 212 in 112 conjugacy classes, 66 normal (38 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×6], C22, C5, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], C10 [×3], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4, C4⋊C4 [×5], C20 [×2], C20 [×6], C2×C10, C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C42.C2, C2×C20, C2×C20 [×2], C2×C20 [×4], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C4⋊Dic3 [×2], C3×C4⋊C4, C5×Dic3 [×4], C60 [×2], C60 [×2], C2×C30, C4×C20, C5×C4⋊C4, C5×C4⋊C4 [×5], C12.3Q8, C10×Dic3 [×2], C10×Dic3 [×2], C2×C60, C2×C60 [×2], C5×C42.C2, Dic3×C20, C5×Dic3⋊C4 [×2], C5×C4⋊Dic3, C5×C4⋊Dic3 [×2], C15×C4⋊C4, C5×C12.3Q8
Quotients: C1, C2 [×7], C22 [×7], C5, S3, Q8 [×2], C23, C10 [×7], D6 [×3], C2×Q8, C4○D4 [×2], C2×C10 [×7], Dic6 [×2], C22×S3, C5×S3, C42.C2, C5×Q8 [×2], C22×C10, C2×Dic6, D4⋊2S3, Q8⋊3S3, S3×C10 [×3], Q8×C10, C5×C4○D4 [×2], C12.3Q8, C5×Dic6 [×2], S3×C2×C10, C5×C42.C2, C10×Dic6, C5×D4⋊2S3, C5×Q8⋊3S3, C5×C12.3Q8
►Regular action on 480 points
Generators in S480
(1 190 429 433 229)(2 191 430 434 230)(3 192 431 435 231)(4 181 432 436 232)(5 182 421 437 233)(6 183 422 438 234)(7 184 423 439 235)(8 185 424 440 236)(9 186 425 441 237)(10 187 426 442 238)(11 188 427 443 239)(12 189 428 444 240)(13 418 204 378 206)(14 419 193 379 207)(15 420 194 380 208)(16 409 195 381 209)(17 410 196 382 210)(18 411 197 383 211)(19 412 198 384 212)(20 413 199 373 213)(21 414 200 374 214)(22 415 201 375 215)(23 416 202 376 216)(24 417 203 377 205)(25 354 41 326 93)(26 355 42 327 94)(27 356 43 328 95)(28 357 44 329 96)(29 358 45 330 85)(30 359 46 331 86)(31 360 47 332 87)(32 349 48 333 88)(33 350 37 334 89)(34 351 38 335 90)(35 352 39 336 91)(36 353 40 325 92)(49 62 396 227 313)(50 63 385 228 314)(51 64 386 217 315)(52 65 387 218 316)(53 66 388 219 317)(54 67 389 220 318)(55 68 390 221 319)(56 69 391 222 320)(57 70 392 223 321)(58 71 393 224 322)(59 72 394 225 323)(60 61 395 226 324)(73 479 372 266 451)(74 480 361 267 452)(75 469 362 268 453)(76 470 363 269 454)(77 471 364 270 455)(78 472 365 271 456)(79 473 366 272 445)(80 474 367 273 446)(81 475 368 274 447)(82 476 369 275 448)(83 477 370 276 449)(84 478 371 265 450)(97 125 138 468 173)(98 126 139 457 174)(99 127 140 458 175)(100 128 141 459 176)(101 129 142 460 177)(102 130 143 461 178)(103 131 144 462 179)(104 132 133 463 180)(105 121 134 464 169)(106 122 135 465 170)(107 123 136 466 171)(108 124 137 467 172)(109 244 167 293 146)(110 245 168 294 147)(111 246 157 295 148)(112 247 158 296 149)(113 248 159 297 150)(114 249 160 298 151)(115 250 161 299 152)(116 251 162 300 153)(117 252 163 289 154)(118 241 164 290 155)(119 242 165 291 156)(120 243 166 292 145)(253 312 343 398 281)(254 301 344 399 282)(255 302 345 400 283)(256 303 346 401 284)(257 304 347 402 285)(258 305 348 403 286)(259 306 337 404 287)(260 307 338 405 288)(261 308 339 406 277)(262 309 340 407 278)(263 310 341 408 279)(264 311 342 397 280)
(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 59 401 83)(2 54 402 78)(3 49 403 73)(4 56 404 80)(5 51 405 75)(6 58 406 82)(7 53 407 77)(8 60 408 84)(9 55 397 79)(10 50 398 74)(11 57 399 81)(12 52 400 76)(13 130 88 146)(14 125 89 153)(15 132 90 148)(16 127 91 155)(17 122 92 150)(18 129 93 145)(19 124 94 152)(20 131 95 147)(21 126 96 154)(22 121 85 149)(23 128 86 156)(24 123 87 151)(25 120 411 142)(26 115 412 137)(27 110 413 144)(28 117 414 139)(29 112 415 134)(30 119 416 141)(31 114 417 136)(32 109 418 143)(33 116 419 138)(34 111 420 133)(35 118 409 140)(36 113 410 135)(37 162 379 173)(38 157 380 180)(39 164 381 175)(40 159 382 170)(41 166 383 177)(42 161 384 172)(43 168 373 179)(44 163 374 174)(45 158 375 169)(46 165 376 176)(47 160 377 171)(48 167 378 178)(61 279 478 185)(62 286 479 192)(63 281 480 187)(64 288 469 182)(65 283 470 189)(66 278 471 184)(67 285 472 191)(68 280 473 186)(69 287 474 181)(70 282 475 188)(71 277 476 183)(72 284 477 190)(97 334 300 207)(98 329 289 214)(99 336 290 209)(100 331 291 216)(101 326 292 211)(102 333 293 206)(103 328 294 213)(104 335 295 208)(105 330 296 215)(106 325 297 210)(107 332 298 205)(108 327 299 212)(193 468 350 251)(194 463 351 246)(195 458 352 241)(196 465 353 248)(197 460 354 243)(198 467 355 250)(199 462 356 245)(200 457 357 252)(201 464 358 247)(202 459 359 242)(203 466 360 249)(204 461 349 244)(217 307 268 437)(218 302 269 444)(219 309 270 439)(220 304 271 434)(221 311 272 441)(222 306 273 436)(223 301 274 443)(224 308 275 438)(225 303 276 433)(226 310 265 440)(227 305 266 435)(228 312 267 442)(229 323 346 449)(230 318 347 456)(231 313 348 451)(232 320 337 446)(233 315 338 453)(234 322 339 448)(235 317 340 455)(236 324 341 450)(237 319 342 445)(238 314 343 452)(239 321 344 447)(240 316 345 454)(253 361 426 385)(254 368 427 392)(255 363 428 387)(256 370 429 394)(257 365 430 389)(258 372 431 396)(259 367 432 391)(260 362 421 386)(261 369 422 393)(262 364 423 388)(263 371 424 395)(264 366 425 390)
(1 246 401 463)(2 251 402 468)(3 244 403 461)(4 249 404 466)(5 242 405 459)(6 247 406 464)(7 252 407 457)(8 245 408 462)(9 250 397 467)(10 243 398 460)(11 248 399 465)(12 241 400 458)(13 221 88 272)(14 226 89 265)(15 219 90 270)(16 224 91 275)(17 217 92 268)(18 222 93 273)(19 227 94 266)(20 220 95 271)(21 225 96 276)(22 218 85 269)(23 223 86 274)(24 228 87 267)(25 446 411 320)(26 451 412 313)(27 456 413 318)(28 449 414 323)(29 454 415 316)(30 447 416 321)(31 452 417 314)(32 445 418 319)(33 450 419 324)(34 455 420 317)(35 448 409 322)(36 453 410 315)(37 478 379 61)(38 471 380 66)(39 476 381 71)(40 469 382 64)(41 474 383 69)(42 479 384 62)(43 472 373 67)(44 477 374 72)(45 470 375 65)(46 475 376 70)(47 480 377 63)(48 473 378 68)(49 355 73 198)(50 360 74 203)(51 353 75 196)(52 358 76 201)(53 351 77 194)(54 356 78 199)(55 349 79 204)(56 354 80 197)(57 359 81 202)(58 352 82 195)(59 357 83 200)(60 350 84 193)(97 430 300 257)(98 423 289 262)(99 428 290 255)(100 421 291 260)(101 426 292 253)(102 431 293 258)(103 424 294 263)(104 429 295 256)(105 422 296 261)(106 427 297 254)(107 432 298 259)(108 425 299 264)(109 348 143 231)(110 341 144 236)(111 346 133 229)(112 339 134 234)(113 344 135 239)(114 337 136 232)(115 342 137 237)(116 347 138 230)(117 340 139 235)(118 345 140 240)(119 338 141 233)(120 343 142 238)(121 438 149 308)(122 443 150 301)(123 436 151 306)(124 441 152 311)(125 434 153 304)(126 439 154 309)(127 444 155 302)(128 437 156 307)(129 442 145 312)(130 435 146 305)(131 440 147 310)(132 433 148 303)(157 284 180 190)(158 277 169 183)(159 282 170 188)(160 287 171 181)(161 280 172 186)(162 285 173 191)(163 278 174 184)(164 283 175 189)(165 288 176 182)(166 281 177 187)(167 286 178 192)(168 279 179 185)(205 385 332 361)(206 390 333 366)(207 395 334 371)(208 388 335 364)(209 393 336 369)(210 386 325 362)(211 391 326 367)(212 396 327 372)(213 389 328 365)(214 394 329 370)(215 387 330 363)(216 392 331 368)
G:=sub<Sym(480)| (1,190,429,433,229)(2,191,430,434,230)(3,192,431,435,231)(4,181,432,436,232)(5,182,421,437,233)(6,183,422,438,234)(7,184,423,439,235)(8,185,424,440,236)(9,186,425,441,237)(10,187,426,442,238)(11,188,427,443,239)(12,189,428,444,240)(13,418,204,378,206)(14,419,193,379,207)(15,420,194,380,208)(16,409,195,381,209)(17,410,196,382,210)(18,411,197,383,211)(19,412,198,384,212)(20,413,199,373,213)(21,414,200,374,214)(22,415,201,375,215)(23,416,202,376,216)(24,417,203,377,205)(25,354,41,326,93)(26,355,42,327,94)(27,356,43,328,95)(28,357,44,329,96)(29,358,45,330,85)(30,359,46,331,86)(31,360,47,332,87)(32,349,48,333,88)(33,350,37,334,89)(34,351,38,335,90)(35,352,39,336,91)(36,353,40,325,92)(49,62,396,227,313)(50,63,385,228,314)(51,64,386,217,315)(52,65,387,218,316)(53,66,388,219,317)(54,67,389,220,318)(55,68,390,221,319)(56,69,391,222,320)(57,70,392,223,321)(58,71,393,224,322)(59,72,394,225,323)(60,61,395,226,324)(73,479,372,266,451)(74,480,361,267,452)(75,469,362,268,453)(76,470,363,269,454)(77,471,364,270,455)(78,472,365,271,456)(79,473,366,272,445)(80,474,367,273,446)(81,475,368,274,447)(82,476,369,275,448)(83,477,370,276,449)(84,478,371,265,450)(97,125,138,468,173)(98,126,139,457,174)(99,127,140,458,175)(100,128,141,459,176)(101,129,142,460,177)(102,130,143,461,178)(103,131,144,462,179)(104,132,133,463,180)(105,121,134,464,169)(106,122,135,465,170)(107,123,136,466,171)(108,124,137,467,172)(109,244,167,293,146)(110,245,168,294,147)(111,246,157,295,148)(112,247,158,296,149)(113,248,159,297,150)(114,249,160,298,151)(115,250,161,299,152)(116,251,162,300,153)(117,252,163,289,154)(118,241,164,290,155)(119,242,165,291,156)(120,243,166,292,145)(253,312,343,398,281)(254,301,344,399,282)(255,302,345,400,283)(256,303,346,401,284)(257,304,347,402,285)(258,305,348,403,286)(259,306,337,404,287)(260,307,338,405,288)(261,308,339,406,277)(262,309,340,407,278)(263,310,341,408,279)(264,311,342,397,280), (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), (1,59,401,83)(2,54,402,78)(3,49,403,73)(4,56,404,80)(5,51,405,75)(6,58,406,82)(7,53,407,77)(8,60,408,84)(9,55,397,79)(10,50,398,74)(11,57,399,81)(12,52,400,76)(13,130,88,146)(14,125,89,153)(15,132,90,148)(16,127,91,155)(17,122,92,150)(18,129,93,145)(19,124,94,152)(20,131,95,147)(21,126,96,154)(22,121,85,149)(23,128,86,156)(24,123,87,151)(25,120,411,142)(26,115,412,137)(27,110,413,144)(28,117,414,139)(29,112,415,134)(30,119,416,141)(31,114,417,136)(32,109,418,143)(33,116,419,138)(34,111,420,133)(35,118,409,140)(36,113,410,135)(37,162,379,173)(38,157,380,180)(39,164,381,175)(40,159,382,170)(41,166,383,177)(42,161,384,172)(43,168,373,179)(44,163,374,174)(45,158,375,169)(46,165,376,176)(47,160,377,171)(48,167,378,178)(61,279,478,185)(62,286,479,192)(63,281,480,187)(64,288,469,182)(65,283,470,189)(66,278,471,184)(67,285,472,191)(68,280,473,186)(69,287,474,181)(70,282,475,188)(71,277,476,183)(72,284,477,190)(97,334,300,207)(98,329,289,214)(99,336,290,209)(100,331,291,216)(101,326,292,211)(102,333,293,206)(103,328,294,213)(104,335,295,208)(105,330,296,215)(106,325,297,210)(107,332,298,205)(108,327,299,212)(193,468,350,251)(194,463,351,246)(195,458,352,241)(196,465,353,248)(197,460,354,243)(198,467,355,250)(199,462,356,245)(200,457,357,252)(201,464,358,247)(202,459,359,242)(203,466,360,249)(204,461,349,244)(217,307,268,437)(218,302,269,444)(219,309,270,439)(220,304,271,434)(221,311,272,441)(222,306,273,436)(223,301,274,443)(224,308,275,438)(225,303,276,433)(226,310,265,440)(227,305,266,435)(228,312,267,442)(229,323,346,449)(230,318,347,456)(231,313,348,451)(232,320,337,446)(233,315,338,453)(234,322,339,448)(235,317,340,455)(236,324,341,450)(237,319,342,445)(238,314,343,452)(239,321,344,447)(240,316,345,454)(253,361,426,385)(254,368,427,392)(255,363,428,387)(256,370,429,394)(257,365,430,389)(258,372,431,396)(259,367,432,391)(260,362,421,386)(261,369,422,393)(262,364,423,388)(263,371,424,395)(264,366,425,390), (1,246,401,463)(2,251,402,468)(3,244,403,461)(4,249,404,466)(5,242,405,459)(6,247,406,464)(7,252,407,457)(8,245,408,462)(9,250,397,467)(10,243,398,460)(11,248,399,465)(12,241,400,458)(13,221,88,272)(14,226,89,265)(15,219,90,270)(16,224,91,275)(17,217,92,268)(18,222,93,273)(19,227,94,266)(20,220,95,271)(21,225,96,276)(22,218,85,269)(23,223,86,274)(24,228,87,267)(25,446,411,320)(26,451,412,313)(27,456,413,318)(28,449,414,323)(29,454,415,316)(30,447,416,321)(31,452,417,314)(32,445,418,319)(33,450,419,324)(34,455,420,317)(35,448,409,322)(36,453,410,315)(37,478,379,61)(38,471,380,66)(39,476,381,71)(40,469,382,64)(41,474,383,69)(42,479,384,62)(43,472,373,67)(44,477,374,72)(45,470,375,65)(46,475,376,70)(47,480,377,63)(48,473,378,68)(49,355,73,198)(50,360,74,203)(51,353,75,196)(52,358,76,201)(53,351,77,194)(54,356,78,199)(55,349,79,204)(56,354,80,197)(57,359,81,202)(58,352,82,195)(59,357,83,200)(60,350,84,193)(97,430,300,257)(98,423,289,262)(99,428,290,255)(100,421,291,260)(101,426,292,253)(102,431,293,258)(103,424,294,263)(104,429,295,256)(105,422,296,261)(106,427,297,254)(107,432,298,259)(108,425,299,264)(109,348,143,231)(110,341,144,236)(111,346,133,229)(112,339,134,234)(113,344,135,239)(114,337,136,232)(115,342,137,237)(116,347,138,230)(117,340,139,235)(118,345,140,240)(119,338,141,233)(120,343,142,238)(121,438,149,308)(122,443,150,301)(123,436,151,306)(124,441,152,311)(125,434,153,304)(126,439,154,309)(127,444,155,302)(128,437,156,307)(129,442,145,312)(130,435,146,305)(131,440,147,310)(132,433,148,303)(157,284,180,190)(158,277,169,183)(159,282,170,188)(160,287,171,181)(161,280,172,186)(162,285,173,191)(163,278,174,184)(164,283,175,189)(165,288,176,182)(166,281,177,187)(167,286,178,192)(168,279,179,185)(205,385,332,361)(206,390,333,366)(207,395,334,371)(208,388,335,364)(209,393,336,369)(210,386,325,362)(211,391,326,367)(212,396,327,372)(213,389,328,365)(214,394,329,370)(215,387,330,363)(216,392,331,368)>;
G:=Group( (1,190,429,433,229)(2,191,430,434,230)(3,192,431,435,231)(4,181,432,436,232)(5,182,421,437,233)(6,183,422,438,234)(7,184,423,439,235)(8,185,424,440,236)(9,186,425,441,237)(10,187,426,442,238)(11,188,427,443,239)(12,189,428,444,240)(13,418,204,378,206)(14,419,193,379,207)(15,420,194,380,208)(16,409,195,381,209)(17,410,196,382,210)(18,411,197,383,211)(19,412,198,384,212)(20,413,199,373,213)(21,414,200,374,214)(22,415,201,375,215)(23,416,202,376,216)(24,417,203,377,205)(25,354,41,326,93)(26,355,42,327,94)(27,356,43,328,95)(28,357,44,329,96)(29,358,45,330,85)(30,359,46,331,86)(31,360,47,332,87)(32,349,48,333,88)(33,350,37,334,89)(34,351,38,335,90)(35,352,39,336,91)(36,353,40,325,92)(49,62,396,227,313)(50,63,385,228,314)(51,64,386,217,315)(52,65,387,218,316)(53,66,388,219,317)(54,67,389,220,318)(55,68,390,221,319)(56,69,391,222,320)(57,70,392,223,321)(58,71,393,224,322)(59,72,394,225,323)(60,61,395,226,324)(73,479,372,266,451)(74,480,361,267,452)(75,469,362,268,453)(76,470,363,269,454)(77,471,364,270,455)(78,472,365,271,456)(79,473,366,272,445)(80,474,367,273,446)(81,475,368,274,447)(82,476,369,275,448)(83,477,370,276,449)(84,478,371,265,450)(97,125,138,468,173)(98,126,139,457,174)(99,127,140,458,175)(100,128,141,459,176)(101,129,142,460,177)(102,130,143,461,178)(103,131,144,462,179)(104,132,133,463,180)(105,121,134,464,169)(106,122,135,465,170)(107,123,136,466,171)(108,124,137,467,172)(109,244,167,293,146)(110,245,168,294,147)(111,246,157,295,148)(112,247,158,296,149)(113,248,159,297,150)(114,249,160,298,151)(115,250,161,299,152)(116,251,162,300,153)(117,252,163,289,154)(118,241,164,290,155)(119,242,165,291,156)(120,243,166,292,145)(253,312,343,398,281)(254,301,344,399,282)(255,302,345,400,283)(256,303,346,401,284)(257,304,347,402,285)(258,305,348,403,286)(259,306,337,404,287)(260,307,338,405,288)(261,308,339,406,277)(262,309,340,407,278)(263,310,341,408,279)(264,311,342,397,280), (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), (1,59,401,83)(2,54,402,78)(3,49,403,73)(4,56,404,80)(5,51,405,75)(6,58,406,82)(7,53,407,77)(8,60,408,84)(9,55,397,79)(10,50,398,74)(11,57,399,81)(12,52,400,76)(13,130,88,146)(14,125,89,153)(15,132,90,148)(16,127,91,155)(17,122,92,150)(18,129,93,145)(19,124,94,152)(20,131,95,147)(21,126,96,154)(22,121,85,149)(23,128,86,156)(24,123,87,151)(25,120,411,142)(26,115,412,137)(27,110,413,144)(28,117,414,139)(29,112,415,134)(30,119,416,141)(31,114,417,136)(32,109,418,143)(33,116,419,138)(34,111,420,133)(35,118,409,140)(36,113,410,135)(37,162,379,173)(38,157,380,180)(39,164,381,175)(40,159,382,170)(41,166,383,177)(42,161,384,172)(43,168,373,179)(44,163,374,174)(45,158,375,169)(46,165,376,176)(47,160,377,171)(48,167,378,178)(61,279,478,185)(62,286,479,192)(63,281,480,187)(64,288,469,182)(65,283,470,189)(66,278,471,184)(67,285,472,191)(68,280,473,186)(69,287,474,181)(70,282,475,188)(71,277,476,183)(72,284,477,190)(97,334,300,207)(98,329,289,214)(99,336,290,209)(100,331,291,216)(101,326,292,211)(102,333,293,206)(103,328,294,213)(104,335,295,208)(105,330,296,215)(106,325,297,210)(107,332,298,205)(108,327,299,212)(193,468,350,251)(194,463,351,246)(195,458,352,241)(196,465,353,248)(197,460,354,243)(198,467,355,250)(199,462,356,245)(200,457,357,252)(201,464,358,247)(202,459,359,242)(203,466,360,249)(204,461,349,244)(217,307,268,437)(218,302,269,444)(219,309,270,439)(220,304,271,434)(221,311,272,441)(222,306,273,436)(223,301,274,443)(224,308,275,438)(225,303,276,433)(226,310,265,440)(227,305,266,435)(228,312,267,442)(229,323,346,449)(230,318,347,456)(231,313,348,451)(232,320,337,446)(233,315,338,453)(234,322,339,448)(235,317,340,455)(236,324,341,450)(237,319,342,445)(238,314,343,452)(239,321,344,447)(240,316,345,454)(253,361,426,385)(254,368,427,392)(255,363,428,387)(256,370,429,394)(257,365,430,389)(258,372,431,396)(259,367,432,391)(260,362,421,386)(261,369,422,393)(262,364,423,388)(263,371,424,395)(264,366,425,390), (1,246,401,463)(2,251,402,468)(3,244,403,461)(4,249,404,466)(5,242,405,459)(6,247,406,464)(7,252,407,457)(8,245,408,462)(9,250,397,467)(10,243,398,460)(11,248,399,465)(12,241,400,458)(13,221,88,272)(14,226,89,265)(15,219,90,270)(16,224,91,275)(17,217,92,268)(18,222,93,273)(19,227,94,266)(20,220,95,271)(21,225,96,276)(22,218,85,269)(23,223,86,274)(24,228,87,267)(25,446,411,320)(26,451,412,313)(27,456,413,318)(28,449,414,323)(29,454,415,316)(30,447,416,321)(31,452,417,314)(32,445,418,319)(33,450,419,324)(34,455,420,317)(35,448,409,322)(36,453,410,315)(37,478,379,61)(38,471,380,66)(39,476,381,71)(40,469,382,64)(41,474,383,69)(42,479,384,62)(43,472,373,67)(44,477,374,72)(45,470,375,65)(46,475,376,70)(47,480,377,63)(48,473,378,68)(49,355,73,198)(50,360,74,203)(51,353,75,196)(52,358,76,201)(53,351,77,194)(54,356,78,199)(55,349,79,204)(56,354,80,197)(57,359,81,202)(58,352,82,195)(59,357,83,200)(60,350,84,193)(97,430,300,257)(98,423,289,262)(99,428,290,255)(100,421,291,260)(101,426,292,253)(102,431,293,258)(103,424,294,263)(104,429,295,256)(105,422,296,261)(106,427,297,254)(107,432,298,259)(108,425,299,264)(109,348,143,231)(110,341,144,236)(111,346,133,229)(112,339,134,234)(113,344,135,239)(114,337,136,232)(115,342,137,237)(116,347,138,230)(117,340,139,235)(118,345,140,240)(119,338,141,233)(120,343,142,238)(121,438,149,308)(122,443,150,301)(123,436,151,306)(124,441,152,311)(125,434,153,304)(126,439,154,309)(127,444,155,302)(128,437,156,307)(129,442,145,312)(130,435,146,305)(131,440,147,310)(132,433,148,303)(157,284,180,190)(158,277,169,183)(159,282,170,188)(160,287,171,181)(161,280,172,186)(162,285,173,191)(163,278,174,184)(164,283,175,189)(165,288,176,182)(166,281,177,187)(167,286,178,192)(168,279,179,185)(205,385,332,361)(206,390,333,366)(207,395,334,371)(208,388,335,364)(209,393,336,369)(210,386,325,362)(211,391,326,367)(212,396,327,372)(213,389,328,365)(214,394,329,370)(215,387,330,363)(216,392,331,368) );
G=PermutationGroup([(1,190,429,433,229),(2,191,430,434,230),(3,192,431,435,231),(4,181,432,436,232),(5,182,421,437,233),(6,183,422,438,234),(7,184,423,439,235),(8,185,424,440,236),(9,186,425,441,237),(10,187,426,442,238),(11,188,427,443,239),(12,189,428,444,240),(13,418,204,378,206),(14,419,193,379,207),(15,420,194,380,208),(16,409,195,381,209),(17,410,196,382,210),(18,411,197,383,211),(19,412,198,384,212),(20,413,199,373,213),(21,414,200,374,214),(22,415,201,375,215),(23,416,202,376,216),(24,417,203,377,205),(25,354,41,326,93),(26,355,42,327,94),(27,356,43,328,95),(28,357,44,329,96),(29,358,45,330,85),(30,359,46,331,86),(31,360,47,332,87),(32,349,48,333,88),(33,350,37,334,89),(34,351,38,335,90),(35,352,39,336,91),(36,353,40,325,92),(49,62,396,227,313),(50,63,385,228,314),(51,64,386,217,315),(52,65,387,218,316),(53,66,388,219,317),(54,67,389,220,318),(55,68,390,221,319),(56,69,391,222,320),(57,70,392,223,321),(58,71,393,224,322),(59,72,394,225,323),(60,61,395,226,324),(73,479,372,266,451),(74,480,361,267,452),(75,469,362,268,453),(76,470,363,269,454),(77,471,364,270,455),(78,472,365,271,456),(79,473,366,272,445),(80,474,367,273,446),(81,475,368,274,447),(82,476,369,275,448),(83,477,370,276,449),(84,478,371,265,450),(97,125,138,468,173),(98,126,139,457,174),(99,127,140,458,175),(100,128,141,459,176),(101,129,142,460,177),(102,130,143,461,178),(103,131,144,462,179),(104,132,133,463,180),(105,121,134,464,169),(106,122,135,465,170),(107,123,136,466,171),(108,124,137,467,172),(109,244,167,293,146),(110,245,168,294,147),(111,246,157,295,148),(112,247,158,296,149),(113,248,159,297,150),(114,249,160,298,151),(115,250,161,299,152),(116,251,162,300,153),(117,252,163,289,154),(118,241,164,290,155),(119,242,165,291,156),(120,243,166,292,145),(253,312,343,398,281),(254,301,344,399,282),(255,302,345,400,283),(256,303,346,401,284),(257,304,347,402,285),(258,305,348,403,286),(259,306,337,404,287),(260,307,338,405,288),(261,308,339,406,277),(262,309,340,407,278),(263,310,341,408,279),(264,311,342,397,280)], [(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)], [(1,59,401,83),(2,54,402,78),(3,49,403,73),(4,56,404,80),(5,51,405,75),(6,58,406,82),(7,53,407,77),(8,60,408,84),(9,55,397,79),(10,50,398,74),(11,57,399,81),(12,52,400,76),(13,130,88,146),(14,125,89,153),(15,132,90,148),(16,127,91,155),(17,122,92,150),(18,129,93,145),(19,124,94,152),(20,131,95,147),(21,126,96,154),(22,121,85,149),(23,128,86,156),(24,123,87,151),(25,120,411,142),(26,115,412,137),(27,110,413,144),(28,117,414,139),(29,112,415,134),(30,119,416,141),(31,114,417,136),(32,109,418,143),(33,116,419,138),(34,111,420,133),(35,118,409,140),(36,113,410,135),(37,162,379,173),(38,157,380,180),(39,164,381,175),(40,159,382,170),(41,166,383,177),(42,161,384,172),(43,168,373,179),(44,163,374,174),(45,158,375,169),(46,165,376,176),(47,160,377,171),(48,167,378,178),(61,279,478,185),(62,286,479,192),(63,281,480,187),(64,288,469,182),(65,283,470,189),(66,278,471,184),(67,285,472,191),(68,280,473,186),(69,287,474,181),(70,282,475,188),(71,277,476,183),(72,284,477,190),(97,334,300,207),(98,329,289,214),(99,336,290,209),(100,331,291,216),(101,326,292,211),(102,333,293,206),(103,328,294,213),(104,335,295,208),(105,330,296,215),(106,325,297,210),(107,332,298,205),(108,327,299,212),(193,468,350,251),(194,463,351,246),(195,458,352,241),(196,465,353,248),(197,460,354,243),(198,467,355,250),(199,462,356,245),(200,457,357,252),(201,464,358,247),(202,459,359,242),(203,466,360,249),(204,461,349,244),(217,307,268,437),(218,302,269,444),(219,309,270,439),(220,304,271,434),(221,311,272,441),(222,306,273,436),(223,301,274,443),(224,308,275,438),(225,303,276,433),(226,310,265,440),(227,305,266,435),(228,312,267,442),(229,323,346,449),(230,318,347,456),(231,313,348,451),(232,320,337,446),(233,315,338,453),(234,322,339,448),(235,317,340,455),(236,324,341,450),(237,319,342,445),(238,314,343,452),(239,321,344,447),(240,316,345,454),(253,361,426,385),(254,368,427,392),(255,363,428,387),(256,370,429,394),(257,365,430,389),(258,372,431,396),(259,367,432,391),(260,362,421,386),(261,369,422,393),(262,364,423,388),(263,371,424,395),(264,366,425,390)], [(1,246,401,463),(2,251,402,468),(3,244,403,461),(4,249,404,466),(5,242,405,459),(6,247,406,464),(7,252,407,457),(8,245,408,462),(9,250,397,467),(10,243,398,460),(11,248,399,465),(12,241,400,458),(13,221,88,272),(14,226,89,265),(15,219,90,270),(16,224,91,275),(17,217,92,268),(18,222,93,273),(19,227,94,266),(20,220,95,271),(21,225,96,276),(22,218,85,269),(23,223,86,274),(24,228,87,267),(25,446,411,320),(26,451,412,313),(27,456,413,318),(28,449,414,323),(29,454,415,316),(30,447,416,321),(31,452,417,314),(32,445,418,319),(33,450,419,324),(34,455,420,317),(35,448,409,322),(36,453,410,315),(37,478,379,61),(38,471,380,66),(39,476,381,71),(40,469,382,64),(41,474,383,69),(42,479,384,62),(43,472,373,67),(44,477,374,72),(45,470,375,65),(46,475,376,70),(47,480,377,63),(48,473,378,68),(49,355,73,198),(50,360,74,203),(51,353,75,196),(52,358,76,201),(53,351,77,194),(54,356,78,199),(55,349,79,204),(56,354,80,197),(57,359,81,202),(58,352,82,195),(59,357,83,200),(60,350,84,193),(97,430,300,257),(98,423,289,262),(99,428,290,255),(100,421,291,260),(101,426,292,253),(102,431,293,258),(103,424,294,263),(104,429,295,256),(105,422,296,261),(106,427,297,254),(107,432,298,259),(108,425,299,264),(109,348,143,231),(110,341,144,236),(111,346,133,229),(112,339,134,234),(113,344,135,239),(114,337,136,232),(115,342,137,237),(116,347,138,230),(117,340,139,235),(118,345,140,240),(119,338,141,233),(120,343,142,238),(121,438,149,308),(122,443,150,301),(123,436,151,306),(124,441,152,311),(125,434,153,304),(126,439,154,309),(127,444,155,302),(128,437,156,307),(129,442,145,312),(130,435,146,305),(131,440,147,310),(132,433,148,303),(157,284,180,190),(158,277,169,183),(159,282,170,188),(160,287,171,181),(161,280,172,186),(162,285,173,191),(163,278,174,184),(164,283,175,189),(165,288,176,182),(166,281,177,187),(167,286,178,192),(168,279,179,185),(205,385,332,361),(206,390,333,366),(207,395,334,371),(208,388,335,364),(209,393,336,369),(210,386,325,362),(211,391,326,367),(212,396,327,372),(213,389,328,365),(214,394,329,370),(215,387,330,363),(216,392,331,368)])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20AN | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 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 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | - | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | Q8 | D6 | C4○D4 | Dic6 | C5×S3 | C5×Q8 | S3×C10 | C5×C4○D4 | C5×Dic6 | D4⋊2S3 | Q8⋊3S3 | C5×D4⋊2S3 | C5×Q8⋊3S3 |
| kernel | C5×C12.3Q8 | Dic3×C20 | C5×Dic3⋊C4 | C5×C4⋊Dic3 | C15×C4⋊C4 | C12.3Q8 | C4×Dic3 | Dic3⋊C4 | C4⋊Dic3 | C3×C4⋊C4 | C5×C4⋊C4 | C60 | C2×C20 | C30 | C20 | C4⋊C4 | C12 | C2×C4 | C6 | C4 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 2 | 3 | 1 | 4 | 4 | 8 | 12 | 4 | 1 | 2 | 3 | 4 | 4 | 4 | 8 | 12 | 16 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C12.3Q8 ►in GL4(𝔽61) generated by
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 25 | 54 |
| 0 | 0 | 11 | 36 |
| 23 | 46 | 0 | 0 |
| 15 | 38 | 0 | 0 |
| 0 | 0 | 27 | 59 |
| 0 | 0 | 59 | 34 |
| 25 | 51 | 0 | 0 |
| 26 | 36 | 0 | 0 |
| 0 | 0 | 30 | 16 |
| 0 | 0 | 1 | 31 |
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,1,0,0,0,0,1],[1,60,0,0,1,0,0,0,0,0,25,11,0,0,54,36],[23,15,0,0,46,38,0,0,0,0,27,59,0,0,59,34],[25,26,0,0,51,36,0,0,0,0,30,1,0,0,16,31] >;
C5×C12.3Q8 in GAP, Magma, Sage, TeX
C_5\times C_{12}._3Q_8 % in TeX
G:=Group("C5xC12.3Q8"); // GroupNames label
G:=SmallGroup(480,769);
// by ID
G=gap.SmallGroup(480,769);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,568,2606,891,226,15686]);
// 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,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations