direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C12.Q8, C60.20Q8, C30.37SD16, C20.17Dic6, C3⋊C8⋊2C20, C15⋊9(C4.Q8), C12.2(C5×Q8), C4.12(S3×C20), C12.2(C2×C20), C30.55(C4⋊C4), C6.6(C5×SD16), C4.2(C5×Dic6), C20.114(C4×S3), C60.172(C2×C4), (C2×C20).345D6, (C2×C30).173D4, C4⋊Dic3.9C10, C10.10(D4.S3), (C2×C60).338C22, C10.23(Dic3⋊C4), C10.10(Q8⋊2S3), (C5×C3⋊C8)⋊14C4, C3⋊1(C5×C4.Q8), C6.3(C5×C4⋊C4), (C2×C3⋊C8).2C10, (C5×C4⋊C4).9S3, C4⋊C4.2(C5×S3), (C3×C4⋊C4).2C10, (C10×C3⋊C8).14C2, (C2×C6).29(C5×D4), (C15×C4⋊C4).16C2, C2.1(C5×D4.S3), (C2×C12).8(C2×C10), (C2×C4).33(S3×C10), C2.4(C5×Dic3⋊C4), C2.1(C5×Q8⋊2S3), (C5×C4⋊Dic3).23C2, C22.13(C5×C3⋊D4), (C2×C10).85(C3⋊D4), SmallGroup(480,127)
Series: Derived ►Chief ►Lower central ►Upper central
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, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C4⋊C4, C2×C8, C20, C20, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C30, C4.Q8, C40, C2×C20, C2×C20, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C60, C60, C2×C30, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C12.Q8, C5×C3⋊C8, C10×Dic3, C2×C60, C2×C60, C5×C4.Q8, C10×C3⋊C8, C5×C4⋊Dic3, C15×C4⋊C4, C5×C12.Q8
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, Q8, C10, D6, C4⋊C4, SD16, C20, C2×C10, Dic6, C4×S3, C3⋊D4, C5×S3, C4.Q8, C2×C20, C5×D4, C5×Q8, Dic3⋊C4, D4.S3, Q8⋊2S3, S3×C10, C5×C4⋊C4, C5×SD16, C12.Q8, C5×Dic6, S3×C20, C5×C3⋊D4, C5×C4.Q8, C5×Dic3⋊C4, C5×D4.S3, C5×Q8⋊2S3, C5×C12.Q8
►Regular action on 480 points
Generators in S480
(1 122 297 308 120)(2 123 298 309 109)(3 124 299 310 110)(4 125 300 311 111)(5 126 289 312 112)(6 127 290 301 113)(7 128 291 302 114)(8 129 292 303 115)(9 130 293 304 116)(10 131 294 305 117)(11 132 295 306 118)(12 121 296 307 119)(13 463 201 232 133)(14 464 202 233 134)(15 465 203 234 135)(16 466 204 235 136)(17 467 193 236 137)(18 468 194 237 138)(19 457 195 238 139)(20 458 196 239 140)(21 459 197 240 141)(22 460 198 229 142)(23 461 199 230 143)(24 462 200 231 144)(25 82 341 221 365)(26 83 342 222 366)(27 84 343 223 367)(28 73 344 224 368)(29 74 345 225 369)(30 75 346 226 370)(31 76 347 227 371)(32 77 348 228 372)(33 78 337 217 361)(34 79 338 218 362)(35 80 339 219 363)(36 81 340 220 364)(37 381 49 357 184)(38 382 50 358 185)(39 383 51 359 186)(40 384 52 360 187)(41 373 53 349 188)(42 374 54 350 189)(43 375 55 351 190)(44 376 56 352 191)(45 377 57 353 192)(46 378 58 354 181)(47 379 59 355 182)(48 380 60 356 183)(61 216 334 399 414)(62 205 335 400 415)(63 206 336 401 416)(64 207 325 402 417)(65 208 326 403 418)(66 209 327 404 419)(67 210 328 405 420)(68 211 329 406 409)(69 212 330 407 410)(70 213 331 408 411)(71 214 332 397 412)(72 215 333 398 413)(85 431 102 324 285)(86 432 103 313 286)(87 421 104 314 287)(88 422 105 315 288)(89 423 106 316 277)(90 424 107 317 278)(91 425 108 318 279)(92 426 97 319 280)(93 427 98 320 281)(94 428 99 321 282)(95 429 100 322 283)(96 430 101 323 284)(145 478 274 433 170)(146 479 275 434 171)(147 480 276 435 172)(148 469 265 436 173)(149 470 266 437 174)(150 471 267 438 175)(151 472 268 439 176)(152 473 269 440 177)(153 474 270 441 178)(154 475 271 442 179)(155 476 272 443 180)(156 477 273 444 169)(157 257 396 249 445)(158 258 385 250 446)(159 259 386 251 447)(160 260 387 252 448)(161 261 388 241 449)(162 262 389 242 450)(163 263 390 243 451)(164 264 391 244 452)(165 253 392 245 453)(166 254 393 246 454)(167 255 394 247 455)(168 256 395 248 456)
(1 25 454 136)(2 137 455 26)(3 27 456 138)(4 139 445 28)(5 29 446 140)(6 141 447 30)(7 31 448 142)(8 143 449 32)(9 33 450 144)(10 133 451 34)(11 35 452 134)(12 135 453 36)(13 163 79 131)(14 132 80 164)(15 165 81 121)(16 122 82 166)(17 167 83 123)(18 124 84 168)(19 157 73 125)(20 126 74 158)(21 159 75 127)(22 128 76 160)(23 161 77 129)(24 130 78 162)(37 443 429 406)(38 407 430 444)(39 433 431 408)(40 397 432 434)(41 435 421 398)(42 399 422 436)(43 437 423 400)(44 401 424 438)(45 439 425 402)(46 403 426 440)(47 441 427 404)(48 405 428 442)(49 155 322 68)(50 69 323 156)(51 145 324 70)(52 71 313 146)(53 147 314 72)(54 61 315 148)(55 149 316 62)(56 63 317 150)(57 151 318 64)(58 65 319 152)(59 153 320 66)(60 67 321 154)(85 331 186 274)(86 275 187 332)(87 333 188 276)(88 265 189 334)(89 335 190 266)(90 267 191 336)(91 325 192 268)(92 269 181 326)(93 327 182 270)(94 271 183 328)(95 329 184 272)(96 273 185 330)(97 177 378 418)(98 419 379 178)(99 179 380 420)(100 409 381 180)(101 169 382 410)(102 411 383 170)(103 171 384 412)(104 413 373 172)(105 173 374 414)(106 415 375 174)(107 175 376 416)(108 417 377 176)(109 236 247 366)(110 367 248 237)(111 238 249 368)(112 369 250 239)(113 240 251 370)(114 371 252 229)(115 230 241 372)(116 361 242 231)(117 232 243 362)(118 363 244 233)(119 234 245 364)(120 365 246 235)(193 394 222 309)(194 310 223 395)(195 396 224 311)(196 312 225 385)(197 386 226 301)(198 302 227 387)(199 388 228 303)(200 304 217 389)(201 390 218 305)(202 306 219 391)(203 392 220 307)(204 308 221 393)(205 351 470 277)(206 278 471 352)(207 353 472 279)(208 280 473 354)(209 355 474 281)(210 282 475 356)(211 357 476 283)(212 284 477 358)(213 359 478 285)(214 286 479 360)(215 349 480 287)(216 288 469 350)(253 340 296 465)(254 466 297 341)(255 342 298 467)(256 468 299 343)(257 344 300 457)(258 458 289 345)(259 346 290 459)(260 460 291 347)(261 348 292 461)(262 462 293 337)(263 338 294 463)(264 464 295 339)
(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)
(1 210 31 288 454 475 142 350)(2 281 143 215 455 355 32 480)(3 208 33 286 456 473 144 360)(4 279 133 213 445 353 34 478)(5 206 35 284 446 471 134 358)(6 277 135 211 447 351 36 476)(7 216 25 282 448 469 136 356)(8 287 137 209 449 349 26 474)(9 214 27 280 450 479 138 354)(10 285 139 207 451 359 28 472)(11 212 29 278 452 477 140 352)(12 283 141 205 453 357 30 470)(13 331 157 192 79 274 125 91)(14 185 126 336 80 96 158 267)(15 329 159 190 81 272 127 89)(16 183 128 334 82 94 160 265)(17 327 161 188 83 270 129 87)(18 181 130 332 84 92 162 275)(19 325 163 186 73 268 131 85)(20 191 132 330 74 90 164 273)(21 335 165 184 75 266 121 95)(22 189 122 328 76 88 166 271)(23 333 167 182 77 276 123 93)(24 187 124 326 78 86 168 269)(37 346 437 296 429 459 400 253)(38 289 401 339 430 258 438 464)(39 344 439 294 431 457 402 263)(40 299 403 337 432 256 440 462)(41 342 441 292 421 467 404 261)(42 297 405 347 422 254 442 460)(43 340 443 290 423 465 406 259)(44 295 407 345 424 264 444 458)(45 338 433 300 425 463 408 257)(46 293 397 343 426 262 434 468)(47 348 435 298 427 461 398 255)(48 291 399 341 428 260 436 466)(49 370 149 119 322 240 62 245)(50 112 63 363 323 250 150 233)(51 368 151 117 324 238 64 243)(52 110 65 361 313 248 152 231)(53 366 153 115 314 236 66 241)(54 120 67 371 315 246 154 229)(55 364 155 113 316 234 68 251)(56 118 69 369 317 244 156 239)(57 362 145 111 318 232 70 249)(58 116 71 367 319 242 146 237)(59 372 147 109 320 230 72 247)(60 114 61 365 321 252 148 235)(97 389 171 194 378 304 412 223)(98 199 413 394 379 228 172 309)(99 387 173 204 380 302 414 221)(100 197 415 392 381 226 174 307)(101 385 175 202 382 312 416 219)(102 195 417 390 383 224 176 305)(103 395 177 200 384 310 418 217)(104 193 419 388 373 222 178 303)(105 393 179 198 374 308 420 227)(106 203 409 386 375 220 180 301)(107 391 169 196 376 306 410 225)(108 201 411 396 377 218 170 311)
G:=sub<Sym(480)| (1,122,297,308,120)(2,123,298,309,109)(3,124,299,310,110)(4,125,300,311,111)(5,126,289,312,112)(6,127,290,301,113)(7,128,291,302,114)(8,129,292,303,115)(9,130,293,304,116)(10,131,294,305,117)(11,132,295,306,118)(12,121,296,307,119)(13,463,201,232,133)(14,464,202,233,134)(15,465,203,234,135)(16,466,204,235,136)(17,467,193,236,137)(18,468,194,237,138)(19,457,195,238,139)(20,458,196,239,140)(21,459,197,240,141)(22,460,198,229,142)(23,461,199,230,143)(24,462,200,231,144)(25,82,341,221,365)(26,83,342,222,366)(27,84,343,223,367)(28,73,344,224,368)(29,74,345,225,369)(30,75,346,226,370)(31,76,347,227,371)(32,77,348,228,372)(33,78,337,217,361)(34,79,338,218,362)(35,80,339,219,363)(36,81,340,220,364)(37,381,49,357,184)(38,382,50,358,185)(39,383,51,359,186)(40,384,52,360,187)(41,373,53,349,188)(42,374,54,350,189)(43,375,55,351,190)(44,376,56,352,191)(45,377,57,353,192)(46,378,58,354,181)(47,379,59,355,182)(48,380,60,356,183)(61,216,334,399,414)(62,205,335,400,415)(63,206,336,401,416)(64,207,325,402,417)(65,208,326,403,418)(66,209,327,404,419)(67,210,328,405,420)(68,211,329,406,409)(69,212,330,407,410)(70,213,331,408,411)(71,214,332,397,412)(72,215,333,398,413)(85,431,102,324,285)(86,432,103,313,286)(87,421,104,314,287)(88,422,105,315,288)(89,423,106,316,277)(90,424,107,317,278)(91,425,108,318,279)(92,426,97,319,280)(93,427,98,320,281)(94,428,99,321,282)(95,429,100,322,283)(96,430,101,323,284)(145,478,274,433,170)(146,479,275,434,171)(147,480,276,435,172)(148,469,265,436,173)(149,470,266,437,174)(150,471,267,438,175)(151,472,268,439,176)(152,473,269,440,177)(153,474,270,441,178)(154,475,271,442,179)(155,476,272,443,180)(156,477,273,444,169)(157,257,396,249,445)(158,258,385,250,446)(159,259,386,251,447)(160,260,387,252,448)(161,261,388,241,449)(162,262,389,242,450)(163,263,390,243,451)(164,264,391,244,452)(165,253,392,245,453)(166,254,393,246,454)(167,255,394,247,455)(168,256,395,248,456), (1,25,454,136)(2,137,455,26)(3,27,456,138)(4,139,445,28)(5,29,446,140)(6,141,447,30)(7,31,448,142)(8,143,449,32)(9,33,450,144)(10,133,451,34)(11,35,452,134)(12,135,453,36)(13,163,79,131)(14,132,80,164)(15,165,81,121)(16,122,82,166)(17,167,83,123)(18,124,84,168)(19,157,73,125)(20,126,74,158)(21,159,75,127)(22,128,76,160)(23,161,77,129)(24,130,78,162)(37,443,429,406)(38,407,430,444)(39,433,431,408)(40,397,432,434)(41,435,421,398)(42,399,422,436)(43,437,423,400)(44,401,424,438)(45,439,425,402)(46,403,426,440)(47,441,427,404)(48,405,428,442)(49,155,322,68)(50,69,323,156)(51,145,324,70)(52,71,313,146)(53,147,314,72)(54,61,315,148)(55,149,316,62)(56,63,317,150)(57,151,318,64)(58,65,319,152)(59,153,320,66)(60,67,321,154)(85,331,186,274)(86,275,187,332)(87,333,188,276)(88,265,189,334)(89,335,190,266)(90,267,191,336)(91,325,192,268)(92,269,181,326)(93,327,182,270)(94,271,183,328)(95,329,184,272)(96,273,185,330)(97,177,378,418)(98,419,379,178)(99,179,380,420)(100,409,381,180)(101,169,382,410)(102,411,383,170)(103,171,384,412)(104,413,373,172)(105,173,374,414)(106,415,375,174)(107,175,376,416)(108,417,377,176)(109,236,247,366)(110,367,248,237)(111,238,249,368)(112,369,250,239)(113,240,251,370)(114,371,252,229)(115,230,241,372)(116,361,242,231)(117,232,243,362)(118,363,244,233)(119,234,245,364)(120,365,246,235)(193,394,222,309)(194,310,223,395)(195,396,224,311)(196,312,225,385)(197,386,226,301)(198,302,227,387)(199,388,228,303)(200,304,217,389)(201,390,218,305)(202,306,219,391)(203,392,220,307)(204,308,221,393)(205,351,470,277)(206,278,471,352)(207,353,472,279)(208,280,473,354)(209,355,474,281)(210,282,475,356)(211,357,476,283)(212,284,477,358)(213,359,478,285)(214,286,479,360)(215,349,480,287)(216,288,469,350)(253,340,296,465)(254,466,297,341)(255,342,298,467)(256,468,299,343)(257,344,300,457)(258,458,289,345)(259,346,290,459)(260,460,291,347)(261,348,292,461)(262,462,293,337)(263,338,294,463)(264,464,295,339), (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), (1,210,31,288,454,475,142,350)(2,281,143,215,455,355,32,480)(3,208,33,286,456,473,144,360)(4,279,133,213,445,353,34,478)(5,206,35,284,446,471,134,358)(6,277,135,211,447,351,36,476)(7,216,25,282,448,469,136,356)(8,287,137,209,449,349,26,474)(9,214,27,280,450,479,138,354)(10,285,139,207,451,359,28,472)(11,212,29,278,452,477,140,352)(12,283,141,205,453,357,30,470)(13,331,157,192,79,274,125,91)(14,185,126,336,80,96,158,267)(15,329,159,190,81,272,127,89)(16,183,128,334,82,94,160,265)(17,327,161,188,83,270,129,87)(18,181,130,332,84,92,162,275)(19,325,163,186,73,268,131,85)(20,191,132,330,74,90,164,273)(21,335,165,184,75,266,121,95)(22,189,122,328,76,88,166,271)(23,333,167,182,77,276,123,93)(24,187,124,326,78,86,168,269)(37,346,437,296,429,459,400,253)(38,289,401,339,430,258,438,464)(39,344,439,294,431,457,402,263)(40,299,403,337,432,256,440,462)(41,342,441,292,421,467,404,261)(42,297,405,347,422,254,442,460)(43,340,443,290,423,465,406,259)(44,295,407,345,424,264,444,458)(45,338,433,300,425,463,408,257)(46,293,397,343,426,262,434,468)(47,348,435,298,427,461,398,255)(48,291,399,341,428,260,436,466)(49,370,149,119,322,240,62,245)(50,112,63,363,323,250,150,233)(51,368,151,117,324,238,64,243)(52,110,65,361,313,248,152,231)(53,366,153,115,314,236,66,241)(54,120,67,371,315,246,154,229)(55,364,155,113,316,234,68,251)(56,118,69,369,317,244,156,239)(57,362,145,111,318,232,70,249)(58,116,71,367,319,242,146,237)(59,372,147,109,320,230,72,247)(60,114,61,365,321,252,148,235)(97,389,171,194,378,304,412,223)(98,199,413,394,379,228,172,309)(99,387,173,204,380,302,414,221)(100,197,415,392,381,226,174,307)(101,385,175,202,382,312,416,219)(102,195,417,390,383,224,176,305)(103,395,177,200,384,310,418,217)(104,193,419,388,373,222,178,303)(105,393,179,198,374,308,420,227)(106,203,409,386,375,220,180,301)(107,391,169,196,376,306,410,225)(108,201,411,396,377,218,170,311)>;
G:=Group( (1,122,297,308,120)(2,123,298,309,109)(3,124,299,310,110)(4,125,300,311,111)(5,126,289,312,112)(6,127,290,301,113)(7,128,291,302,114)(8,129,292,303,115)(9,130,293,304,116)(10,131,294,305,117)(11,132,295,306,118)(12,121,296,307,119)(13,463,201,232,133)(14,464,202,233,134)(15,465,203,234,135)(16,466,204,235,136)(17,467,193,236,137)(18,468,194,237,138)(19,457,195,238,139)(20,458,196,239,140)(21,459,197,240,141)(22,460,198,229,142)(23,461,199,230,143)(24,462,200,231,144)(25,82,341,221,365)(26,83,342,222,366)(27,84,343,223,367)(28,73,344,224,368)(29,74,345,225,369)(30,75,346,226,370)(31,76,347,227,371)(32,77,348,228,372)(33,78,337,217,361)(34,79,338,218,362)(35,80,339,219,363)(36,81,340,220,364)(37,381,49,357,184)(38,382,50,358,185)(39,383,51,359,186)(40,384,52,360,187)(41,373,53,349,188)(42,374,54,350,189)(43,375,55,351,190)(44,376,56,352,191)(45,377,57,353,192)(46,378,58,354,181)(47,379,59,355,182)(48,380,60,356,183)(61,216,334,399,414)(62,205,335,400,415)(63,206,336,401,416)(64,207,325,402,417)(65,208,326,403,418)(66,209,327,404,419)(67,210,328,405,420)(68,211,329,406,409)(69,212,330,407,410)(70,213,331,408,411)(71,214,332,397,412)(72,215,333,398,413)(85,431,102,324,285)(86,432,103,313,286)(87,421,104,314,287)(88,422,105,315,288)(89,423,106,316,277)(90,424,107,317,278)(91,425,108,318,279)(92,426,97,319,280)(93,427,98,320,281)(94,428,99,321,282)(95,429,100,322,283)(96,430,101,323,284)(145,478,274,433,170)(146,479,275,434,171)(147,480,276,435,172)(148,469,265,436,173)(149,470,266,437,174)(150,471,267,438,175)(151,472,268,439,176)(152,473,269,440,177)(153,474,270,441,178)(154,475,271,442,179)(155,476,272,443,180)(156,477,273,444,169)(157,257,396,249,445)(158,258,385,250,446)(159,259,386,251,447)(160,260,387,252,448)(161,261,388,241,449)(162,262,389,242,450)(163,263,390,243,451)(164,264,391,244,452)(165,253,392,245,453)(166,254,393,246,454)(167,255,394,247,455)(168,256,395,248,456), (1,25,454,136)(2,137,455,26)(3,27,456,138)(4,139,445,28)(5,29,446,140)(6,141,447,30)(7,31,448,142)(8,143,449,32)(9,33,450,144)(10,133,451,34)(11,35,452,134)(12,135,453,36)(13,163,79,131)(14,132,80,164)(15,165,81,121)(16,122,82,166)(17,167,83,123)(18,124,84,168)(19,157,73,125)(20,126,74,158)(21,159,75,127)(22,128,76,160)(23,161,77,129)(24,130,78,162)(37,443,429,406)(38,407,430,444)(39,433,431,408)(40,397,432,434)(41,435,421,398)(42,399,422,436)(43,437,423,400)(44,401,424,438)(45,439,425,402)(46,403,426,440)(47,441,427,404)(48,405,428,442)(49,155,322,68)(50,69,323,156)(51,145,324,70)(52,71,313,146)(53,147,314,72)(54,61,315,148)(55,149,316,62)(56,63,317,150)(57,151,318,64)(58,65,319,152)(59,153,320,66)(60,67,321,154)(85,331,186,274)(86,275,187,332)(87,333,188,276)(88,265,189,334)(89,335,190,266)(90,267,191,336)(91,325,192,268)(92,269,181,326)(93,327,182,270)(94,271,183,328)(95,329,184,272)(96,273,185,330)(97,177,378,418)(98,419,379,178)(99,179,380,420)(100,409,381,180)(101,169,382,410)(102,411,383,170)(103,171,384,412)(104,413,373,172)(105,173,374,414)(106,415,375,174)(107,175,376,416)(108,417,377,176)(109,236,247,366)(110,367,248,237)(111,238,249,368)(112,369,250,239)(113,240,251,370)(114,371,252,229)(115,230,241,372)(116,361,242,231)(117,232,243,362)(118,363,244,233)(119,234,245,364)(120,365,246,235)(193,394,222,309)(194,310,223,395)(195,396,224,311)(196,312,225,385)(197,386,226,301)(198,302,227,387)(199,388,228,303)(200,304,217,389)(201,390,218,305)(202,306,219,391)(203,392,220,307)(204,308,221,393)(205,351,470,277)(206,278,471,352)(207,353,472,279)(208,280,473,354)(209,355,474,281)(210,282,475,356)(211,357,476,283)(212,284,477,358)(213,359,478,285)(214,286,479,360)(215,349,480,287)(216,288,469,350)(253,340,296,465)(254,466,297,341)(255,342,298,467)(256,468,299,343)(257,344,300,457)(258,458,289,345)(259,346,290,459)(260,460,291,347)(261,348,292,461)(262,462,293,337)(263,338,294,463)(264,464,295,339), (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), (1,210,31,288,454,475,142,350)(2,281,143,215,455,355,32,480)(3,208,33,286,456,473,144,360)(4,279,133,213,445,353,34,478)(5,206,35,284,446,471,134,358)(6,277,135,211,447,351,36,476)(7,216,25,282,448,469,136,356)(8,287,137,209,449,349,26,474)(9,214,27,280,450,479,138,354)(10,285,139,207,451,359,28,472)(11,212,29,278,452,477,140,352)(12,283,141,205,453,357,30,470)(13,331,157,192,79,274,125,91)(14,185,126,336,80,96,158,267)(15,329,159,190,81,272,127,89)(16,183,128,334,82,94,160,265)(17,327,161,188,83,270,129,87)(18,181,130,332,84,92,162,275)(19,325,163,186,73,268,131,85)(20,191,132,330,74,90,164,273)(21,335,165,184,75,266,121,95)(22,189,122,328,76,88,166,271)(23,333,167,182,77,276,123,93)(24,187,124,326,78,86,168,269)(37,346,437,296,429,459,400,253)(38,289,401,339,430,258,438,464)(39,344,439,294,431,457,402,263)(40,299,403,337,432,256,440,462)(41,342,441,292,421,467,404,261)(42,297,405,347,422,254,442,460)(43,340,443,290,423,465,406,259)(44,295,407,345,424,264,444,458)(45,338,433,300,425,463,408,257)(46,293,397,343,426,262,434,468)(47,348,435,298,427,461,398,255)(48,291,399,341,428,260,436,466)(49,370,149,119,322,240,62,245)(50,112,63,363,323,250,150,233)(51,368,151,117,324,238,64,243)(52,110,65,361,313,248,152,231)(53,366,153,115,314,236,66,241)(54,120,67,371,315,246,154,229)(55,364,155,113,316,234,68,251)(56,118,69,369,317,244,156,239)(57,362,145,111,318,232,70,249)(58,116,71,367,319,242,146,237)(59,372,147,109,320,230,72,247)(60,114,61,365,321,252,148,235)(97,389,171,194,378,304,412,223)(98,199,413,394,379,228,172,309)(99,387,173,204,380,302,414,221)(100,197,415,392,381,226,174,307)(101,385,175,202,382,312,416,219)(102,195,417,390,383,224,176,305)(103,395,177,200,384,310,418,217)(104,193,419,388,373,222,178,303)(105,393,179,198,374,308,420,227)(106,203,409,386,375,220,180,301)(107,391,169,196,376,306,410,225)(108,201,411,396,377,218,170,311) );
G=PermutationGroup([[(1,122,297,308,120),(2,123,298,309,109),(3,124,299,310,110),(4,125,300,311,111),(5,126,289,312,112),(6,127,290,301,113),(7,128,291,302,114),(8,129,292,303,115),(9,130,293,304,116),(10,131,294,305,117),(11,132,295,306,118),(12,121,296,307,119),(13,463,201,232,133),(14,464,202,233,134),(15,465,203,234,135),(16,466,204,235,136),(17,467,193,236,137),(18,468,194,237,138),(19,457,195,238,139),(20,458,196,239,140),(21,459,197,240,141),(22,460,198,229,142),(23,461,199,230,143),(24,462,200,231,144),(25,82,341,221,365),(26,83,342,222,366),(27,84,343,223,367),(28,73,344,224,368),(29,74,345,225,369),(30,75,346,226,370),(31,76,347,227,371),(32,77,348,228,372),(33,78,337,217,361),(34,79,338,218,362),(35,80,339,219,363),(36,81,340,220,364),(37,381,49,357,184),(38,382,50,358,185),(39,383,51,359,186),(40,384,52,360,187),(41,373,53,349,188),(42,374,54,350,189),(43,375,55,351,190),(44,376,56,352,191),(45,377,57,353,192),(46,378,58,354,181),(47,379,59,355,182),(48,380,60,356,183),(61,216,334,399,414),(62,205,335,400,415),(63,206,336,401,416),(64,207,325,402,417),(65,208,326,403,418),(66,209,327,404,419),(67,210,328,405,420),(68,211,329,406,409),(69,212,330,407,410),(70,213,331,408,411),(71,214,332,397,412),(72,215,333,398,413),(85,431,102,324,285),(86,432,103,313,286),(87,421,104,314,287),(88,422,105,315,288),(89,423,106,316,277),(90,424,107,317,278),(91,425,108,318,279),(92,426,97,319,280),(93,427,98,320,281),(94,428,99,321,282),(95,429,100,322,283),(96,430,101,323,284),(145,478,274,433,170),(146,479,275,434,171),(147,480,276,435,172),(148,469,265,436,173),(149,470,266,437,174),(150,471,267,438,175),(151,472,268,439,176),(152,473,269,440,177),(153,474,270,441,178),(154,475,271,442,179),(155,476,272,443,180),(156,477,273,444,169),(157,257,396,249,445),(158,258,385,250,446),(159,259,386,251,447),(160,260,387,252,448),(161,261,388,241,449),(162,262,389,242,450),(163,263,390,243,451),(164,264,391,244,452),(165,253,392,245,453),(166,254,393,246,454),(167,255,394,247,455),(168,256,395,248,456)], [(1,25,454,136),(2,137,455,26),(3,27,456,138),(4,139,445,28),(5,29,446,140),(6,141,447,30),(7,31,448,142),(8,143,449,32),(9,33,450,144),(10,133,451,34),(11,35,452,134),(12,135,453,36),(13,163,79,131),(14,132,80,164),(15,165,81,121),(16,122,82,166),(17,167,83,123),(18,124,84,168),(19,157,73,125),(20,126,74,158),(21,159,75,127),(22,128,76,160),(23,161,77,129),(24,130,78,162),(37,443,429,406),(38,407,430,444),(39,433,431,408),(40,397,432,434),(41,435,421,398),(42,399,422,436),(43,437,423,400),(44,401,424,438),(45,439,425,402),(46,403,426,440),(47,441,427,404),(48,405,428,442),(49,155,322,68),(50,69,323,156),(51,145,324,70),(52,71,313,146),(53,147,314,72),(54,61,315,148),(55,149,316,62),(56,63,317,150),(57,151,318,64),(58,65,319,152),(59,153,320,66),(60,67,321,154),(85,331,186,274),(86,275,187,332),(87,333,188,276),(88,265,189,334),(89,335,190,266),(90,267,191,336),(91,325,192,268),(92,269,181,326),(93,327,182,270),(94,271,183,328),(95,329,184,272),(96,273,185,330),(97,177,378,418),(98,419,379,178),(99,179,380,420),(100,409,381,180),(101,169,382,410),(102,411,383,170),(103,171,384,412),(104,413,373,172),(105,173,374,414),(106,415,375,174),(107,175,376,416),(108,417,377,176),(109,236,247,366),(110,367,248,237),(111,238,249,368),(112,369,250,239),(113,240,251,370),(114,371,252,229),(115,230,241,372),(116,361,242,231),(117,232,243,362),(118,363,244,233),(119,234,245,364),(120,365,246,235),(193,394,222,309),(194,310,223,395),(195,396,224,311),(196,312,225,385),(197,386,226,301),(198,302,227,387),(199,388,228,303),(200,304,217,389),(201,390,218,305),(202,306,219,391),(203,392,220,307),(204,308,221,393),(205,351,470,277),(206,278,471,352),(207,353,472,279),(208,280,473,354),(209,355,474,281),(210,282,475,356),(211,357,476,283),(212,284,477,358),(213,359,478,285),(214,286,479,360),(215,349,480,287),(216,288,469,350),(253,340,296,465),(254,466,297,341),(255,342,298,467),(256,468,299,343),(257,344,300,457),(258,458,289,345),(259,346,290,459),(260,460,291,347),(261,348,292,461),(262,462,293,337),(263,338,294,463),(264,464,295,339)], [(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)], [(1,210,31,288,454,475,142,350),(2,281,143,215,455,355,32,480),(3,208,33,286,456,473,144,360),(4,279,133,213,445,353,34,478),(5,206,35,284,446,471,134,358),(6,277,135,211,447,351,36,476),(7,216,25,282,448,469,136,356),(8,287,137,209,449,349,26,474),(9,214,27,280,450,479,138,354),(10,285,139,207,451,359,28,472),(11,212,29,278,452,477,140,352),(12,283,141,205,453,357,30,470),(13,331,157,192,79,274,125,91),(14,185,126,336,80,96,158,267),(15,329,159,190,81,272,127,89),(16,183,128,334,82,94,160,265),(17,327,161,188,83,270,129,87),(18,181,130,332,84,92,162,275),(19,325,163,186,73,268,131,85),(20,191,132,330,74,90,164,273),(21,335,165,184,75,266,121,95),(22,189,122,328,76,88,166,271),(23,333,167,182,77,276,123,93),(24,187,124,326,78,86,168,269),(37,346,437,296,429,459,400,253),(38,289,401,339,430,258,438,464),(39,344,439,294,431,457,402,263),(40,299,403,337,432,256,440,462),(41,342,441,292,421,467,404,261),(42,297,405,347,422,254,442,460),(43,340,443,290,423,465,406,259),(44,295,407,345,424,264,444,458),(45,338,433,300,425,463,408,257),(46,293,397,343,426,262,434,468),(47,348,435,298,427,461,398,255),(48,291,399,341,428,260,436,466),(49,370,149,119,322,240,62,245),(50,112,63,363,323,250,150,233),(51,368,151,117,324,238,64,243),(52,110,65,361,313,248,152,231),(53,366,153,115,314,236,66,241),(54,120,67,371,315,246,154,229),(55,364,155,113,316,234,68,251),(56,118,69,369,317,244,156,239),(57,362,145,111,318,232,70,249),(58,116,71,367,319,242,146,237),(59,372,147,109,320,230,72,247),(60,114,61,365,321,252,148,235),(97,389,171,194,378,304,412,223),(98,199,413,394,379,228,172,309),(99,387,173,204,380,302,414,221),(100,197,415,392,381,226,174,307),(101,385,175,202,382,312,416,219),(102,195,417,390,383,224,176,305),(103,395,177,200,384,310,418,217),(104,193,419,388,373,222,178,303),(105,393,179,198,374,308,420,227),(106,203,409,386,375,220,180,301),(107,391,169,196,376,306,410,225),(108,201,411,396,377,218,170,311)]])
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