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## G = C5×C12.Q8order 480 = 25·3·5

### Direct product of C5 and C12.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C12.Q8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C60 — C10×C3⋊C8 — C5×C12.Q8
 Lower central C3 — C6 — C12 — C5×C12.Q8
 Upper central C1 — C2×C10 — C2×C20 — C5×C4⋊C4

Generators and relations for C5×C12.Q8
G = < a,b,c,d | a5=b4=c12=1, d2=bc6, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 148 in 72 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3, C12 [×2], C12, C2×C6, C15, C4⋊C4, C4⋊C4, C2×C8, C20 [×2], C20 [×2], C2×C10, C3⋊C8 [×2], C2×Dic3, C2×C12, C2×C12, C30 [×3], C4.Q8, C40 [×2], C2×C20, C2×C20 [×2], C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C60 [×2], C60, C2×C30, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C12.Q8, C5×C3⋊C8 [×2], C10×Dic3, C2×C60, C2×C60, C5×C4.Q8, C10×C3⋊C8, C5×C4⋊Dic3, C15×C4⋊C4, C5×C12.Q8
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4, Q8, C10 [×3], D6, C4⋊C4, SD16 [×2], C20 [×2], C2×C10, Dic6, C4×S3, C3⋊D4, C5×S3, C4.Q8, C2×C20, C5×D4, C5×Q8, Dic3⋊C4, D4.S3, Q82S3, S3×C10, C5×C4⋊C4, C5×SD16 [×2], C12.Q8, C5×Dic6, S3×C20, C5×C3⋊D4, C5×C4.Q8, C5×Dic3⋊C4, C5×D4.S3, C5×Q82S3, C5×C12.Q8

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

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 10A ··· 10L 12A ··· 12F 15A 15B 15C 15D 20A ··· 20H 20I ··· 20P 20Q ··· 20X 30A ··· 30L 40A ··· 40P 60A ··· 60X order 1 2 2 2 3 4 4 4 4 4 4 5 5 5 5 6 6 6 8 8 8 8 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 2 2 2 4 4 12 12 1 1 1 1 2 2 2 6 6 6 6 1 ··· 1 4 ··· 4 2 2 2 2 2 ··· 2 4 ··· 4 12 ··· 12 2 ··· 2 6 ··· 6 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 2 2 2 2 2 2 4 4 4 4 type + + + + + - + + - - + image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 S3 Q8 D4 D6 SD16 Dic6 C4×S3 C3⋊D4 C5×S3 C5×Q8 C5×D4 S3×C10 C5×SD16 C5×Dic6 S3×C20 C5×C3⋊D4 D4.S3 Q8⋊2S3 C5×D4.S3 C5×Q8⋊2S3 kernel C5×C12.Q8 C10×C3⋊C8 C5×C4⋊Dic3 C15×C4⋊C4 C5×C3⋊C8 C12.Q8 C2×C3⋊C8 C4⋊Dic3 C3×C4⋊C4 C3⋊C8 C5×C4⋊C4 C60 C2×C30 C2×C20 C30 C20 C20 C2×C10 C4⋊C4 C12 C2×C6 C2×C4 C6 C4 C4 C22 C10 C10 C2 C2 # reps 1 1 1 1 4 4 4 4 4 16 1 1 1 1 4 2 2 2 4 4 4 4 16 8 8 8 1 1 4 4

Matrix representation of C5×C12.Q8 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 98 0 0 0 0 98
,
 240 0 0 0 0 240 0 0 0 0 240 238 0 0 81 1
,
 177 177 0 0 64 0 0 0 0 0 22 82 0 0 191 219
,
 175 62 0 0 128 66 0 0 0 0 0 57 0 0 148 203
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,98,0,0,0,0,98],[240,0,0,0,0,240,0,0,0,0,240,81,0,0,238,1],[177,64,0,0,177,0,0,0,0,0,22,191,0,0,82,219],[175,128,0,0,62,66,0,0,0,0,0,148,0,0,57,203] >;

C5×C12.Q8 in GAP, Magma, Sage, TeX

C_5\times C_{12}.Q_8
% in TeX

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

G:=SmallGroup(480,127);
// by ID

G=gap.SmallGroup(480,127);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,280,1709,148,2111,102,15686]);
// Polycyclic

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

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