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G = C5×C12.6Q8order 480 = 25·3·5

Direct product of C5 and C12.6Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C12.6Q8, C60.32Q8, C20.26Dic6, C6.3(Q8×C10), C12.6(C5×Q8), (C4×C20).11S3, (C4×C12).3C10, (C4×C60).13C2, C42.5(C5×S3), C30.84(C2×Q8), C4.6(C5×Dic6), (C2×C20).427D6, C4⋊Dic3.5C10, C2.5(C10×Dic6), Dic3⋊C4.1C10, C10.43(C2×Dic6), C1515(C42.C2), C30.198(C4○D4), (C2×C30).390C23, (C2×C60).450C22, C10.109(C4○D12), (C10×Dic3).135C22, C6.2(C5×C4○D4), C31(C5×C42.C2), C2.6(C5×C4○D12), (C2×C4).63(S3×C10), C22.35(S3×C2×C10), (C2×C12).90(C2×C10), (C5×Dic3⋊C4).1C2, (C5×C4⋊Dic3).19C2, (C2×C6).11(C22×C10), (C2×Dic3).2(C2×C10), (C2×C10).324(C22×S3), SmallGroup(480,749)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×C12.6Q8
C1C3C6C2×C6C2×C30C10×Dic3C5×Dic3⋊C4 — C5×C12.6Q8
C3C2×C6 — C5×C12.6Q8
C1C2×C10C4×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 [×2], C3, C4 [×2], C4 [×6], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], C10, C10 [×2], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4 [×6], C20 [×2], C20 [×6], C2×C10, C2×Dic3 [×4], C2×C12, C2×C12 [×2], C30, C30 [×2], C42.C2, C2×C20, C2×C20 [×2], C2×C20 [×4], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C4×C12, C5×Dic3 [×4], C60 [×2], C60 [×2], C2×C30, C4×C20, C5×C4⋊C4 [×6], C12.6Q8, C10×Dic3 [×4], C2×C60, C2×C60 [×2], C5×C42.C2, C5×Dic3⋊C4 [×4], C5×C4⋊Dic3 [×2], C4×C60, C5×C12.6Q8
Quotients: C1, C2 [×7], C22 [×7], C5, S3, Q8 [×2], C23, C10 [×7], D6 [×3], C2×Q8, C4○D4 [×2], C2×C10 [×7], Dic6 [×2], C22×S3, C5×S3, C42.C2, C5×Q8 [×2], C22×C10, C2×Dic6, C4○D12 [×2], S3×C10 [×3], Q8×C10, C5×C4○D4 [×2], C12.6Q8, C5×Dic6 [×2], S3×C2×C10, C5×C42.C2, C10×Dic6, C5×C4○D12 [×2], C5×C12.6Q8

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

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

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

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

150 conjugacy classes

class 1 2A2B2C 3 4A···4F4G4H4I4J5A5B5C5D6A6B6C10A···10L12A···12L15A15B15C15D20A···20X20Y···20AN30A···30L60A···60AV
order122234···44444555566610···1012···121515151520···2020···2030···3060···60
size111122···21212121211112221···12···222222···212···122···22···2

150 irreducible representations

dim11111111222222222222
type+++++-+-
imageC1C2C2C2C5C10C10C10S3Q8D6C4○D4Dic6C5×S3C5×Q8C4○D12S3×C10C5×C4○D4C5×Dic6C5×C4○D12
kernelC5×C12.6Q8C5×Dic3⋊C4C5×C4⋊Dic3C4×C60C12.6Q8Dic3⋊C4C4⋊Dic3C4×C12C4×C20C60C2×C20C30C20C42C12C10C2×C4C6C4C2
# reps1421416841234448812161632

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

58000
05800
0090
0009
,
60000
06000
003823
003815
,
50000
01100
005243
00189
,
0100
60000
002636
001035
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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