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

Generators and relations for C5×C12⋊Q8
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=c-1 >

Subgroups: 276 in 136 conjugacy classes, 74 normal (38 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×8], C22, C5, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10 [×3], Dic3 [×4], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4, C4⋊C4 [×3], C2×Q8 [×2], C20 [×2], C20 [×8], C2×C10, Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C30 [×3], C4⋊Q8, C2×C20, C2×C20 [×2], C2×C20 [×4], C5×Q8 [×4], C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, C2×Dic6 [×2], C5×Dic3 [×4], C5×Dic3 [×2], C60 [×2], C60 [×2], C2×C30, C4×C20, C5×C4⋊C4, C5×C4⋊C4 [×3], Q8×C10 [×2], C12⋊Q8, C5×Dic6 [×4], C10×Dic3 [×2], C10×Dic3 [×2], C2×C60, C2×C60 [×2], C5×C4⋊Q8, Dic3×C20, C5×Dic3⋊C4 [×2], C5×C4⋊Dic3, C15×C4⋊C4, C10×Dic6 [×2], C5×C12⋊Q8
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], Q8 [×4], C23, C10 [×7], D6 [×3], C2×D4, C2×Q8 [×2], C2×C10 [×7], Dic6 [×2], C22×S3, C5×S3, C4⋊Q8, C5×D4 [×2], C5×Q8 [×4], C22×C10, C2×Dic6, S3×D4, S3×Q8, S3×C10 [×3], D4×C10, Q8×C10 [×2], C12⋊Q8, C5×Dic6 [×2], S3×C2×C10, C5×C4⋊Q8, C10×Dic6, C5×S3×D4, C5×S3×Q8, C5×C12⋊Q8

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

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

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

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

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 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - - + - + - image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D4 Q8 Q8 D6 Dic6 C5×S3 C5×D4 C5×Q8 C5×Q8 S3×C10 C5×Dic6 S3×D4 S3×Q8 C5×S3×D4 C5×S3×Q8 kernel C5×C12⋊Q8 Dic3×C20 C5×Dic3⋊C4 C5×C4⋊Dic3 C15×C4⋊C4 C10×Dic6 C12⋊Q8 C4×Dic3 Dic3⋊C4 C4⋊Dic3 C3×C4⋊C4 C2×Dic6 C5×C4⋊C4 C5×Dic3 C5×Dic3 C60 C2×C20 C20 C4⋊C4 Dic3 Dic3 C12 C2×C4 C4 C10 C10 C2 C2 # reps 1 1 2 1 1 2 4 4 8 4 4 8 1 2 2 2 3 4 4 8 8 8 12 16 1 1 4 4

Matrix representation of C5×C12⋊Q8 in GL6(𝔽61)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 34 0 0 0 0 0 0 34 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 1 60 0 0 0 0 0 0 31 40 0 0 0 0 40 30
,
 0 1 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 60 0
,
 53 22 0 0 0 0 22 8 0 0 0 0 0 0 13 39 0 0 0 0 52 48 0 0 0 0 0 0 30 21 0 0 0 0 21 31

G:=sub<GL(6,GF(61))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,34,0,0,0,0,0,0,34,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,60,60,0,0,0,0,0,0,31,40,0,0,0,0,40,30],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[53,22,0,0,0,0,22,8,0,0,0,0,0,0,13,52,0,0,0,0,39,48,0,0,0,0,0,0,30,21,0,0,0,0,21,31] >;

C5×C12⋊Q8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(480,767);
// by ID

G=gap.SmallGroup(480,767);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,568,926,891,226,15686]);
// 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=c^-1>;
// generators/relations

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