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## G = C5×C12.6Q8order 480 = 25·3·5

### Direct product of C5 and C12.6Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C5×C12.6Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — C5×Dic3⋊C4 — C5×C12.6Q8
 Lower central C3 — C2×C6 — C5×C12.6Q8
 Upper central C1 — C2×C10 — C4×C20

Generators and relations for C5×C12.6Q8
G = < a,b,c,d | a5=b12=c4=1, d2=b6c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b6c-1 >

Subgroups: 212 in 112 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, C10, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C20, C20, C2×C10, C2×Dic3, C2×C12, C2×C12, C30, C30, C42.C2, C2×C20, C2×C20, C2×C20, Dic3⋊C4, C4⋊Dic3, C4×C12, C5×Dic3, C60, C60, C2×C30, C4×C20, C5×C4⋊C4, C12.6Q8, C10×Dic3, C2×C60, C2×C60, C5×C42.C2, C5×Dic3⋊C4, C5×C4⋊Dic3, C4×C60, C5×C12.6Q8
Quotients: C1, C2, C22, C5, S3, Q8, C23, C10, D6, C2×Q8, C4○D4, C2×C10, Dic6, C22×S3, C5×S3, C42.C2, C5×Q8, C22×C10, C2×Dic6, C4○D12, S3×C10, Q8×C10, C5×C4○D4, C12.6Q8, C5×Dic6, S3×C2×C10, C5×C42.C2, C10×Dic6, C5×C4○D12, C5×C12.6Q8

Smallest permutation representation of C5×C12.6Q8
Regular action on 480 points
Generators in S480
(1 155 104 342 373)(2 156 105 343 374)(3 145 106 344 375)(4 146 107 345 376)(5 147 108 346 377)(6 148 97 347 378)(7 149 98 348 379)(8 150 99 337 380)(9 151 100 338 381)(10 152 101 339 382)(11 153 102 340 383)(12 154 103 341 384)(13 142 424 92 313)(14 143 425 93 314)(15 144 426 94 315)(16 133 427 95 316)(17 134 428 96 317)(18 135 429 85 318)(19 136 430 86 319)(20 137 431 87 320)(21 138 432 88 321)(22 139 421 89 322)(23 140 422 90 323)(24 141 423 91 324)(25 265 74 175 456)(26 266 75 176 445)(27 267 76 177 446)(28 268 77 178 447)(29 269 78 179 448)(30 270 79 180 449)(31 271 80 169 450)(32 272 81 170 451)(33 273 82 171 452)(34 274 83 172 453)(35 275 84 173 454)(36 276 73 174 455)(37 470 227 331 190)(38 471 228 332 191)(39 472 217 333 192)(40 473 218 334 181)(41 474 219 335 182)(42 475 220 336 183)(43 476 221 325 184)(44 477 222 326 185)(45 478 223 327 186)(46 479 224 328 187)(47 480 225 329 188)(48 469 226 330 189)(49 124 388 292 353)(50 125 389 293 354)(51 126 390 294 355)(52 127 391 295 356)(53 128 392 296 357)(54 129 393 297 358)(55 130 394 298 359)(56 131 395 299 360)(57 132 396 300 349)(58 121 385 289 350)(59 122 386 290 351)(60 123 387 291 352)(61 413 459 263 363)(62 414 460 264 364)(63 415 461 253 365)(64 416 462 254 366)(65 417 463 255 367)(66 418 464 256 368)(67 419 465 257 369)(68 420 466 258 370)(69 409 467 259 371)(70 410 468 260 372)(71 411 457 261 361)(72 412 458 262 362)(109 196 166 301 216)(110 197 167 302 205)(111 198 168 303 206)(112 199 157 304 207)(113 200 158 305 208)(114 201 159 306 209)(115 202 160 307 210)(116 203 161 308 211)(117 204 162 309 212)(118 193 163 310 213)(119 194 164 311 214)(120 195 165 312 215)(229 243 280 434 400)(230 244 281 435 401)(231 245 282 436 402)(232 246 283 437 403)(233 247 284 438 404)(234 248 285 439 405)(235 249 286 440 406)(236 250 287 441 407)(237 251 288 442 408)(238 252 277 443 397)(239 241 278 444 398)(240 242 279 433 399)
(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)
(1 466 44 450)(2 467 45 451)(3 468 46 452)(4 457 47 453)(5 458 48 454)(6 459 37 455)(7 460 38 456)(8 461 39 445)(9 462 40 446)(10 463 41 447)(11 464 42 448)(12 465 43 449)(13 215 398 125)(14 216 399 126)(15 205 400 127)(16 206 401 128)(17 207 402 129)(18 208 403 130)(19 209 404 131)(20 210 405 132)(21 211 406 121)(22 212 407 122)(23 213 408 123)(24 214 397 124)(25 149 264 471)(26 150 253 472)(27 151 254 473)(28 152 255 474)(29 153 256 475)(30 154 257 476)(31 155 258 477)(32 156 259 478)(33 145 260 479)(34 146 261 480)(35 147 262 469)(36 148 263 470)(49 324 311 443)(50 313 312 444)(51 314 301 433)(52 315 302 434)(53 316 303 435)(54 317 304 436)(55 318 305 437)(56 319 306 438)(57 320 307 439)(58 321 308 440)(59 322 309 441)(60 323 310 442)(61 331 73 347)(62 332 74 348)(63 333 75 337)(64 334 76 338)(65 335 77 339)(66 336 78 340)(67 325 79 341)(68 326 80 342)(69 327 81 343)(70 328 82 344)(71 329 83 345)(72 330 84 346)(85 158 283 359)(86 159 284 360)(87 160 285 349)(88 161 286 350)(89 162 287 351)(90 163 288 352)(91 164 277 353)(92 165 278 354)(93 166 279 355)(94 167 280 356)(95 168 281 357)(96 157 282 358)(97 363 227 276)(98 364 228 265)(99 365 217 266)(100 366 218 267)(101 367 219 268)(102 368 220 269)(103 369 221 270)(104 370 222 271)(105 371 223 272)(106 372 224 273)(107 361 225 274)(108 362 226 275)(109 240 390 143)(110 229 391 144)(111 230 392 133)(112 231 393 134)(113 232 394 135)(114 233 395 136)(115 234 396 137)(116 235 385 138)(117 236 386 139)(118 237 387 140)(119 238 388 141)(120 239 389 142)(169 373 420 185)(170 374 409 186)(171 375 410 187)(172 376 411 188)(173 377 412 189)(174 378 413 190)(175 379 414 191)(176 380 415 192)(177 381 416 181)(178 382 417 182)(179 383 418 183)(180 384 419 184)(193 251 291 422)(194 252 292 423)(195 241 293 424)(196 242 294 425)(197 243 295 426)(198 244 296 427)(199 245 297 428)(200 246 298 429)(201 247 299 430)(202 248 300 431)(203 249 289 432)(204 250 290 421)
(1 207 38 123)(2 206 39 122)(3 205 40 121)(4 216 41 132)(5 215 42 131)(6 214 43 130)(7 213 44 129)(8 212 45 128)(9 211 46 127)(10 210 47 126)(11 209 48 125)(12 208 37 124)(13 454 404 464)(14 453 405 463)(15 452 406 462)(16 451 407 461)(17 450 408 460)(18 449 397 459)(19 448 398 458)(20 447 399 457)(21 446 400 468)(22 445 401 467)(23 456 402 466)(24 455 403 465)(25 231 258 140)(26 230 259 139)(27 229 260 138)(28 240 261 137)(29 239 262 136)(30 238 263 135)(31 237 264 134)(32 236 253 133)(33 235 254 144)(34 234 255 143)(35 233 256 142)(36 232 257 141)(49 384 305 190)(50 383 306 189)(51 382 307 188)(52 381 308 187)(53 380 309 186)(54 379 310 185)(55 378 311 184)(56 377 312 183)(57 376 301 182)(58 375 302 181)(59 374 303 192)(60 373 304 191)(61 85 79 277)(62 96 80 288)(63 95 81 287)(64 94 82 286)(65 93 83 285)(66 92 84 284)(67 91 73 283)(68 90 74 282)(69 89 75 281)(70 88 76 280)(71 87 77 279)(72 86 78 278)(97 194 221 298)(98 193 222 297)(99 204 223 296)(100 203 224 295)(101 202 225 294)(102 201 226 293)(103 200 227 292)(104 199 228 291)(105 198 217 290)(106 197 218 289)(107 196 219 300)(108 195 220 299)(109 474 396 146)(110 473 385 145)(111 472 386 156)(112 471 387 155)(113 470 388 154)(114 469 389 153)(115 480 390 152)(116 479 391 151)(117 478 392 150)(118 477 393 149)(119 476 394 148)(120 475 395 147)(157 332 352 342)(158 331 353 341)(159 330 354 340)(160 329 355 339)(161 328 356 338)(162 327 357 337)(163 326 358 348)(164 325 359 347)(165 336 360 346)(166 335 349 345)(167 334 350 344)(168 333 351 343)(169 442 414 317)(170 441 415 316)(171 440 416 315)(172 439 417 314)(173 438 418 313)(174 437 419 324)(175 436 420 323)(176 435 409 322)(177 434 410 321)(178 433 411 320)(179 444 412 319)(180 443 413 318)(241 362 430 269)(242 361 431 268)(243 372 432 267)(244 371 421 266)(245 370 422 265)(246 369 423 276)(247 368 424 275)(248 367 425 274)(249 366 426 273)(250 365 427 272)(251 364 428 271)(252 363 429 270)

G:=sub<Sym(480)| (1,155,104,342,373)(2,156,105,343,374)(3,145,106,344,375)(4,146,107,345,376)(5,147,108,346,377)(6,148,97,347,378)(7,149,98,348,379)(8,150,99,337,380)(9,151,100,338,381)(10,152,101,339,382)(11,153,102,340,383)(12,154,103,341,384)(13,142,424,92,313)(14,143,425,93,314)(15,144,426,94,315)(16,133,427,95,316)(17,134,428,96,317)(18,135,429,85,318)(19,136,430,86,319)(20,137,431,87,320)(21,138,432,88,321)(22,139,421,89,322)(23,140,422,90,323)(24,141,423,91,324)(25,265,74,175,456)(26,266,75,176,445)(27,267,76,177,446)(28,268,77,178,447)(29,269,78,179,448)(30,270,79,180,449)(31,271,80,169,450)(32,272,81,170,451)(33,273,82,171,452)(34,274,83,172,453)(35,275,84,173,454)(36,276,73,174,455)(37,470,227,331,190)(38,471,228,332,191)(39,472,217,333,192)(40,473,218,334,181)(41,474,219,335,182)(42,475,220,336,183)(43,476,221,325,184)(44,477,222,326,185)(45,478,223,327,186)(46,479,224,328,187)(47,480,225,329,188)(48,469,226,330,189)(49,124,388,292,353)(50,125,389,293,354)(51,126,390,294,355)(52,127,391,295,356)(53,128,392,296,357)(54,129,393,297,358)(55,130,394,298,359)(56,131,395,299,360)(57,132,396,300,349)(58,121,385,289,350)(59,122,386,290,351)(60,123,387,291,352)(61,413,459,263,363)(62,414,460,264,364)(63,415,461,253,365)(64,416,462,254,366)(65,417,463,255,367)(66,418,464,256,368)(67,419,465,257,369)(68,420,466,258,370)(69,409,467,259,371)(70,410,468,260,372)(71,411,457,261,361)(72,412,458,262,362)(109,196,166,301,216)(110,197,167,302,205)(111,198,168,303,206)(112,199,157,304,207)(113,200,158,305,208)(114,201,159,306,209)(115,202,160,307,210)(116,203,161,308,211)(117,204,162,309,212)(118,193,163,310,213)(119,194,164,311,214)(120,195,165,312,215)(229,243,280,434,400)(230,244,281,435,401)(231,245,282,436,402)(232,246,283,437,403)(233,247,284,438,404)(234,248,285,439,405)(235,249,286,440,406)(236,250,287,441,407)(237,251,288,442,408)(238,252,277,443,397)(239,241,278,444,398)(240,242,279,433,399), (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), (1,466,44,450)(2,467,45,451)(3,468,46,452)(4,457,47,453)(5,458,48,454)(6,459,37,455)(7,460,38,456)(8,461,39,445)(9,462,40,446)(10,463,41,447)(11,464,42,448)(12,465,43,449)(13,215,398,125)(14,216,399,126)(15,205,400,127)(16,206,401,128)(17,207,402,129)(18,208,403,130)(19,209,404,131)(20,210,405,132)(21,211,406,121)(22,212,407,122)(23,213,408,123)(24,214,397,124)(25,149,264,471)(26,150,253,472)(27,151,254,473)(28,152,255,474)(29,153,256,475)(30,154,257,476)(31,155,258,477)(32,156,259,478)(33,145,260,479)(34,146,261,480)(35,147,262,469)(36,148,263,470)(49,324,311,443)(50,313,312,444)(51,314,301,433)(52,315,302,434)(53,316,303,435)(54,317,304,436)(55,318,305,437)(56,319,306,438)(57,320,307,439)(58,321,308,440)(59,322,309,441)(60,323,310,442)(61,331,73,347)(62,332,74,348)(63,333,75,337)(64,334,76,338)(65,335,77,339)(66,336,78,340)(67,325,79,341)(68,326,80,342)(69,327,81,343)(70,328,82,344)(71,329,83,345)(72,330,84,346)(85,158,283,359)(86,159,284,360)(87,160,285,349)(88,161,286,350)(89,162,287,351)(90,163,288,352)(91,164,277,353)(92,165,278,354)(93,166,279,355)(94,167,280,356)(95,168,281,357)(96,157,282,358)(97,363,227,276)(98,364,228,265)(99,365,217,266)(100,366,218,267)(101,367,219,268)(102,368,220,269)(103,369,221,270)(104,370,222,271)(105,371,223,272)(106,372,224,273)(107,361,225,274)(108,362,226,275)(109,240,390,143)(110,229,391,144)(111,230,392,133)(112,231,393,134)(113,232,394,135)(114,233,395,136)(115,234,396,137)(116,235,385,138)(117,236,386,139)(118,237,387,140)(119,238,388,141)(120,239,389,142)(169,373,420,185)(170,374,409,186)(171,375,410,187)(172,376,411,188)(173,377,412,189)(174,378,413,190)(175,379,414,191)(176,380,415,192)(177,381,416,181)(178,382,417,182)(179,383,418,183)(180,384,419,184)(193,251,291,422)(194,252,292,423)(195,241,293,424)(196,242,294,425)(197,243,295,426)(198,244,296,427)(199,245,297,428)(200,246,298,429)(201,247,299,430)(202,248,300,431)(203,249,289,432)(204,250,290,421), (1,207,38,123)(2,206,39,122)(3,205,40,121)(4,216,41,132)(5,215,42,131)(6,214,43,130)(7,213,44,129)(8,212,45,128)(9,211,46,127)(10,210,47,126)(11,209,48,125)(12,208,37,124)(13,454,404,464)(14,453,405,463)(15,452,406,462)(16,451,407,461)(17,450,408,460)(18,449,397,459)(19,448,398,458)(20,447,399,457)(21,446,400,468)(22,445,401,467)(23,456,402,466)(24,455,403,465)(25,231,258,140)(26,230,259,139)(27,229,260,138)(28,240,261,137)(29,239,262,136)(30,238,263,135)(31,237,264,134)(32,236,253,133)(33,235,254,144)(34,234,255,143)(35,233,256,142)(36,232,257,141)(49,384,305,190)(50,383,306,189)(51,382,307,188)(52,381,308,187)(53,380,309,186)(54,379,310,185)(55,378,311,184)(56,377,312,183)(57,376,301,182)(58,375,302,181)(59,374,303,192)(60,373,304,191)(61,85,79,277)(62,96,80,288)(63,95,81,287)(64,94,82,286)(65,93,83,285)(66,92,84,284)(67,91,73,283)(68,90,74,282)(69,89,75,281)(70,88,76,280)(71,87,77,279)(72,86,78,278)(97,194,221,298)(98,193,222,297)(99,204,223,296)(100,203,224,295)(101,202,225,294)(102,201,226,293)(103,200,227,292)(104,199,228,291)(105,198,217,290)(106,197,218,289)(107,196,219,300)(108,195,220,299)(109,474,396,146)(110,473,385,145)(111,472,386,156)(112,471,387,155)(113,470,388,154)(114,469,389,153)(115,480,390,152)(116,479,391,151)(117,478,392,150)(118,477,393,149)(119,476,394,148)(120,475,395,147)(157,332,352,342)(158,331,353,341)(159,330,354,340)(160,329,355,339)(161,328,356,338)(162,327,357,337)(163,326,358,348)(164,325,359,347)(165,336,360,346)(166,335,349,345)(167,334,350,344)(168,333,351,343)(169,442,414,317)(170,441,415,316)(171,440,416,315)(172,439,417,314)(173,438,418,313)(174,437,419,324)(175,436,420,323)(176,435,409,322)(177,434,410,321)(178,433,411,320)(179,444,412,319)(180,443,413,318)(241,362,430,269)(242,361,431,268)(243,372,432,267)(244,371,421,266)(245,370,422,265)(246,369,423,276)(247,368,424,275)(248,367,425,274)(249,366,426,273)(250,365,427,272)(251,364,428,271)(252,363,429,270)>;

G:=Group( (1,155,104,342,373)(2,156,105,343,374)(3,145,106,344,375)(4,146,107,345,376)(5,147,108,346,377)(6,148,97,347,378)(7,149,98,348,379)(8,150,99,337,380)(9,151,100,338,381)(10,152,101,339,382)(11,153,102,340,383)(12,154,103,341,384)(13,142,424,92,313)(14,143,425,93,314)(15,144,426,94,315)(16,133,427,95,316)(17,134,428,96,317)(18,135,429,85,318)(19,136,430,86,319)(20,137,431,87,320)(21,138,432,88,321)(22,139,421,89,322)(23,140,422,90,323)(24,141,423,91,324)(25,265,74,175,456)(26,266,75,176,445)(27,267,76,177,446)(28,268,77,178,447)(29,269,78,179,448)(30,270,79,180,449)(31,271,80,169,450)(32,272,81,170,451)(33,273,82,171,452)(34,274,83,172,453)(35,275,84,173,454)(36,276,73,174,455)(37,470,227,331,190)(38,471,228,332,191)(39,472,217,333,192)(40,473,218,334,181)(41,474,219,335,182)(42,475,220,336,183)(43,476,221,325,184)(44,477,222,326,185)(45,478,223,327,186)(46,479,224,328,187)(47,480,225,329,188)(48,469,226,330,189)(49,124,388,292,353)(50,125,389,293,354)(51,126,390,294,355)(52,127,391,295,356)(53,128,392,296,357)(54,129,393,297,358)(55,130,394,298,359)(56,131,395,299,360)(57,132,396,300,349)(58,121,385,289,350)(59,122,386,290,351)(60,123,387,291,352)(61,413,459,263,363)(62,414,460,264,364)(63,415,461,253,365)(64,416,462,254,366)(65,417,463,255,367)(66,418,464,256,368)(67,419,465,257,369)(68,420,466,258,370)(69,409,467,259,371)(70,410,468,260,372)(71,411,457,261,361)(72,412,458,262,362)(109,196,166,301,216)(110,197,167,302,205)(111,198,168,303,206)(112,199,157,304,207)(113,200,158,305,208)(114,201,159,306,209)(115,202,160,307,210)(116,203,161,308,211)(117,204,162,309,212)(118,193,163,310,213)(119,194,164,311,214)(120,195,165,312,215)(229,243,280,434,400)(230,244,281,435,401)(231,245,282,436,402)(232,246,283,437,403)(233,247,284,438,404)(234,248,285,439,405)(235,249,286,440,406)(236,250,287,441,407)(237,251,288,442,408)(238,252,277,443,397)(239,241,278,444,398)(240,242,279,433,399), (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), (1,466,44,450)(2,467,45,451)(3,468,46,452)(4,457,47,453)(5,458,48,454)(6,459,37,455)(7,460,38,456)(8,461,39,445)(9,462,40,446)(10,463,41,447)(11,464,42,448)(12,465,43,449)(13,215,398,125)(14,216,399,126)(15,205,400,127)(16,206,401,128)(17,207,402,129)(18,208,403,130)(19,209,404,131)(20,210,405,132)(21,211,406,121)(22,212,407,122)(23,213,408,123)(24,214,397,124)(25,149,264,471)(26,150,253,472)(27,151,254,473)(28,152,255,474)(29,153,256,475)(30,154,257,476)(31,155,258,477)(32,156,259,478)(33,145,260,479)(34,146,261,480)(35,147,262,469)(36,148,263,470)(49,324,311,443)(50,313,312,444)(51,314,301,433)(52,315,302,434)(53,316,303,435)(54,317,304,436)(55,318,305,437)(56,319,306,438)(57,320,307,439)(58,321,308,440)(59,322,309,441)(60,323,310,442)(61,331,73,347)(62,332,74,348)(63,333,75,337)(64,334,76,338)(65,335,77,339)(66,336,78,340)(67,325,79,341)(68,326,80,342)(69,327,81,343)(70,328,82,344)(71,329,83,345)(72,330,84,346)(85,158,283,359)(86,159,284,360)(87,160,285,349)(88,161,286,350)(89,162,287,351)(90,163,288,352)(91,164,277,353)(92,165,278,354)(93,166,279,355)(94,167,280,356)(95,168,281,357)(96,157,282,358)(97,363,227,276)(98,364,228,265)(99,365,217,266)(100,366,218,267)(101,367,219,268)(102,368,220,269)(103,369,221,270)(104,370,222,271)(105,371,223,272)(106,372,224,273)(107,361,225,274)(108,362,226,275)(109,240,390,143)(110,229,391,144)(111,230,392,133)(112,231,393,134)(113,232,394,135)(114,233,395,136)(115,234,396,137)(116,235,385,138)(117,236,386,139)(118,237,387,140)(119,238,388,141)(120,239,389,142)(169,373,420,185)(170,374,409,186)(171,375,410,187)(172,376,411,188)(173,377,412,189)(174,378,413,190)(175,379,414,191)(176,380,415,192)(177,381,416,181)(178,382,417,182)(179,383,418,183)(180,384,419,184)(193,251,291,422)(194,252,292,423)(195,241,293,424)(196,242,294,425)(197,243,295,426)(198,244,296,427)(199,245,297,428)(200,246,298,429)(201,247,299,430)(202,248,300,431)(203,249,289,432)(204,250,290,421), (1,207,38,123)(2,206,39,122)(3,205,40,121)(4,216,41,132)(5,215,42,131)(6,214,43,130)(7,213,44,129)(8,212,45,128)(9,211,46,127)(10,210,47,126)(11,209,48,125)(12,208,37,124)(13,454,404,464)(14,453,405,463)(15,452,406,462)(16,451,407,461)(17,450,408,460)(18,449,397,459)(19,448,398,458)(20,447,399,457)(21,446,400,468)(22,445,401,467)(23,456,402,466)(24,455,403,465)(25,231,258,140)(26,230,259,139)(27,229,260,138)(28,240,261,137)(29,239,262,136)(30,238,263,135)(31,237,264,134)(32,236,253,133)(33,235,254,144)(34,234,255,143)(35,233,256,142)(36,232,257,141)(49,384,305,190)(50,383,306,189)(51,382,307,188)(52,381,308,187)(53,380,309,186)(54,379,310,185)(55,378,311,184)(56,377,312,183)(57,376,301,182)(58,375,302,181)(59,374,303,192)(60,373,304,191)(61,85,79,277)(62,96,80,288)(63,95,81,287)(64,94,82,286)(65,93,83,285)(66,92,84,284)(67,91,73,283)(68,90,74,282)(69,89,75,281)(70,88,76,280)(71,87,77,279)(72,86,78,278)(97,194,221,298)(98,193,222,297)(99,204,223,296)(100,203,224,295)(101,202,225,294)(102,201,226,293)(103,200,227,292)(104,199,228,291)(105,198,217,290)(106,197,218,289)(107,196,219,300)(108,195,220,299)(109,474,396,146)(110,473,385,145)(111,472,386,156)(112,471,387,155)(113,470,388,154)(114,469,389,153)(115,480,390,152)(116,479,391,151)(117,478,392,150)(118,477,393,149)(119,476,394,148)(120,475,395,147)(157,332,352,342)(158,331,353,341)(159,330,354,340)(160,329,355,339)(161,328,356,338)(162,327,357,337)(163,326,358,348)(164,325,359,347)(165,336,360,346)(166,335,349,345)(167,334,350,344)(168,333,351,343)(169,442,414,317)(170,441,415,316)(171,440,416,315)(172,439,417,314)(173,438,418,313)(174,437,419,324)(175,436,420,323)(176,435,409,322)(177,434,410,321)(178,433,411,320)(179,444,412,319)(180,443,413,318)(241,362,430,269)(242,361,431,268)(243,372,432,267)(244,371,421,266)(245,370,422,265)(246,369,423,276)(247,368,424,275)(248,367,425,274)(249,366,426,273)(250,365,427,272)(251,364,428,271)(252,363,429,270) );

G=PermutationGroup([[(1,155,104,342,373),(2,156,105,343,374),(3,145,106,344,375),(4,146,107,345,376),(5,147,108,346,377),(6,148,97,347,378),(7,149,98,348,379),(8,150,99,337,380),(9,151,100,338,381),(10,152,101,339,382),(11,153,102,340,383),(12,154,103,341,384),(13,142,424,92,313),(14,143,425,93,314),(15,144,426,94,315),(16,133,427,95,316),(17,134,428,96,317),(18,135,429,85,318),(19,136,430,86,319),(20,137,431,87,320),(21,138,432,88,321),(22,139,421,89,322),(23,140,422,90,323),(24,141,423,91,324),(25,265,74,175,456),(26,266,75,176,445),(27,267,76,177,446),(28,268,77,178,447),(29,269,78,179,448),(30,270,79,180,449),(31,271,80,169,450),(32,272,81,170,451),(33,273,82,171,452),(34,274,83,172,453),(35,275,84,173,454),(36,276,73,174,455),(37,470,227,331,190),(38,471,228,332,191),(39,472,217,333,192),(40,473,218,334,181),(41,474,219,335,182),(42,475,220,336,183),(43,476,221,325,184),(44,477,222,326,185),(45,478,223,327,186),(46,479,224,328,187),(47,480,225,329,188),(48,469,226,330,189),(49,124,388,292,353),(50,125,389,293,354),(51,126,390,294,355),(52,127,391,295,356),(53,128,392,296,357),(54,129,393,297,358),(55,130,394,298,359),(56,131,395,299,360),(57,132,396,300,349),(58,121,385,289,350),(59,122,386,290,351),(60,123,387,291,352),(61,413,459,263,363),(62,414,460,264,364),(63,415,461,253,365),(64,416,462,254,366),(65,417,463,255,367),(66,418,464,256,368),(67,419,465,257,369),(68,420,466,258,370),(69,409,467,259,371),(70,410,468,260,372),(71,411,457,261,361),(72,412,458,262,362),(109,196,166,301,216),(110,197,167,302,205),(111,198,168,303,206),(112,199,157,304,207),(113,200,158,305,208),(114,201,159,306,209),(115,202,160,307,210),(116,203,161,308,211),(117,204,162,309,212),(118,193,163,310,213),(119,194,164,311,214),(120,195,165,312,215),(229,243,280,434,400),(230,244,281,435,401),(231,245,282,436,402),(232,246,283,437,403),(233,247,284,438,404),(234,248,285,439,405),(235,249,286,440,406),(236,250,287,441,407),(237,251,288,442,408),(238,252,277,443,397),(239,241,278,444,398),(240,242,279,433,399)], [(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)], [(1,466,44,450),(2,467,45,451),(3,468,46,452),(4,457,47,453),(5,458,48,454),(6,459,37,455),(7,460,38,456),(8,461,39,445),(9,462,40,446),(10,463,41,447),(11,464,42,448),(12,465,43,449),(13,215,398,125),(14,216,399,126),(15,205,400,127),(16,206,401,128),(17,207,402,129),(18,208,403,130),(19,209,404,131),(20,210,405,132),(21,211,406,121),(22,212,407,122),(23,213,408,123),(24,214,397,124),(25,149,264,471),(26,150,253,472),(27,151,254,473),(28,152,255,474),(29,153,256,475),(30,154,257,476),(31,155,258,477),(32,156,259,478),(33,145,260,479),(34,146,261,480),(35,147,262,469),(36,148,263,470),(49,324,311,443),(50,313,312,444),(51,314,301,433),(52,315,302,434),(53,316,303,435),(54,317,304,436),(55,318,305,437),(56,319,306,438),(57,320,307,439),(58,321,308,440),(59,322,309,441),(60,323,310,442),(61,331,73,347),(62,332,74,348),(63,333,75,337),(64,334,76,338),(65,335,77,339),(66,336,78,340),(67,325,79,341),(68,326,80,342),(69,327,81,343),(70,328,82,344),(71,329,83,345),(72,330,84,346),(85,158,283,359),(86,159,284,360),(87,160,285,349),(88,161,286,350),(89,162,287,351),(90,163,288,352),(91,164,277,353),(92,165,278,354),(93,166,279,355),(94,167,280,356),(95,168,281,357),(96,157,282,358),(97,363,227,276),(98,364,228,265),(99,365,217,266),(100,366,218,267),(101,367,219,268),(102,368,220,269),(103,369,221,270),(104,370,222,271),(105,371,223,272),(106,372,224,273),(107,361,225,274),(108,362,226,275),(109,240,390,143),(110,229,391,144),(111,230,392,133),(112,231,393,134),(113,232,394,135),(114,233,395,136),(115,234,396,137),(116,235,385,138),(117,236,386,139),(118,237,387,140),(119,238,388,141),(120,239,389,142),(169,373,420,185),(170,374,409,186),(171,375,410,187),(172,376,411,188),(173,377,412,189),(174,378,413,190),(175,379,414,191),(176,380,415,192),(177,381,416,181),(178,382,417,182),(179,383,418,183),(180,384,419,184),(193,251,291,422),(194,252,292,423),(195,241,293,424),(196,242,294,425),(197,243,295,426),(198,244,296,427),(199,245,297,428),(200,246,298,429),(201,247,299,430),(202,248,300,431),(203,249,289,432),(204,250,290,421)], [(1,207,38,123),(2,206,39,122),(3,205,40,121),(4,216,41,132),(5,215,42,131),(6,214,43,130),(7,213,44,129),(8,212,45,128),(9,211,46,127),(10,210,47,126),(11,209,48,125),(12,208,37,124),(13,454,404,464),(14,453,405,463),(15,452,406,462),(16,451,407,461),(17,450,408,460),(18,449,397,459),(19,448,398,458),(20,447,399,457),(21,446,400,468),(22,445,401,467),(23,456,402,466),(24,455,403,465),(25,231,258,140),(26,230,259,139),(27,229,260,138),(28,240,261,137),(29,239,262,136),(30,238,263,135),(31,237,264,134),(32,236,253,133),(33,235,254,144),(34,234,255,143),(35,233,256,142),(36,232,257,141),(49,384,305,190),(50,383,306,189),(51,382,307,188),(52,381,308,187),(53,380,309,186),(54,379,310,185),(55,378,311,184),(56,377,312,183),(57,376,301,182),(58,375,302,181),(59,374,303,192),(60,373,304,191),(61,85,79,277),(62,96,80,288),(63,95,81,287),(64,94,82,286),(65,93,83,285),(66,92,84,284),(67,91,73,283),(68,90,74,282),(69,89,75,281),(70,88,76,280),(71,87,77,279),(72,86,78,278),(97,194,221,298),(98,193,222,297),(99,204,223,296),(100,203,224,295),(101,202,225,294),(102,201,226,293),(103,200,227,292),(104,199,228,291),(105,198,217,290),(106,197,218,289),(107,196,219,300),(108,195,220,299),(109,474,396,146),(110,473,385,145),(111,472,386,156),(112,471,387,155),(113,470,388,154),(114,469,389,153),(115,480,390,152),(116,479,391,151),(117,478,392,150),(118,477,393,149),(119,476,394,148),(120,475,395,147),(157,332,352,342),(158,331,353,341),(159,330,354,340),(160,329,355,339),(161,328,356,338),(162,327,357,337),(163,326,358,348),(164,325,359,347),(165,336,360,346),(166,335,349,345),(167,334,350,344),(168,333,351,343),(169,442,414,317),(170,441,415,316),(171,440,416,315),(172,439,417,314),(173,438,418,313),(174,437,419,324),(175,436,420,323),(176,435,409,322),(177,434,410,321),(178,433,411,320),(179,444,412,319),(180,443,413,318),(241,362,430,269),(242,361,431,268),(243,372,432,267),(244,371,421,266),(245,370,422,265),(246,369,423,276),(247,368,424,275),(248,367,425,274),(249,366,426,273),(250,365,427,272),(251,364,428,271),(252,363,429,270)]])

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 Q8 D6 C4○D4 Dic6 C5×S3 C5×Q8 C4○D12 S3×C10 C5×C4○D4 C5×Dic6 C5×C4○D12 kernel C5×C12.6Q8 C5×Dic3⋊C4 C5×C4⋊Dic3 C4×C60 C12.6Q8 Dic3⋊C4 C4⋊Dic3 C4×C12 C4×C20 C60 C2×C20 C30 C20 C42 C12 C10 C2×C4 C6 C4 C2 # reps 1 4 2 1 4 16 8 4 1 2 3 4 4 4 8 8 12 16 16 32

Matrix representation of C5×C12.6Q8 in GL4(𝔽61) generated by

 58 0 0 0 0 58 0 0 0 0 9 0 0 0 0 9
,
 60 0 0 0 0 60 0 0 0 0 38 23 0 0 38 15
,
 50 0 0 0 0 11 0 0 0 0 52 43 0 0 18 9
,
 0 1 0 0 60 0 0 0 0 0 26 36 0 0 10 35
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,9,0,0,0,0,9],[60,0,0,0,0,60,0,0,0,0,38,38,0,0,23,15],[50,0,0,0,0,11,0,0,0,0,52,18,0,0,43,9],[0,60,0,0,1,0,0,0,0,0,26,10,0,0,36,35] >;

C5×C12.6Q8 in GAP, Magma, Sage, TeX

C_5\times C_{12}._6Q_8
% in TeX

G:=Group("C5xC12.6Q8");
// GroupNames label

G:=SmallGroup(480,749);
// by ID

G=gap.SmallGroup(480,749);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,1149,288,926,436,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^12=c^4=1,d^2=b^6*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=b^6*c^-1>;
// generators/relations

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