Copied to
clipboard

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, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C4⋊C4, C2×Q8, C20, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4⋊Q8, C2×C20, C2×C20, C2×C20, C5×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C5×Dic3, C5×Dic3, C60, C60, C2×C30, C4×C20, C5×C4⋊C4, C5×C4⋊C4, Q8×C10, C12⋊Q8, C5×Dic6, C10×Dic3, C10×Dic3, C2×C60, C2×C60, C5×C4⋊Q8, Dic3×C20, C5×Dic3⋊C4, C5×C4⋊Dic3, C15×C4⋊C4, C10×Dic6, C5×C12⋊Q8
Quotients: C1, C2, C22, C5, S3, D4, Q8, C23, C10, D6, C2×D4, C2×Q8, C2×C10, Dic6, C22×S3, C5×S3, C4⋊Q8, C5×D4, C5×Q8, C22×C10, C2×Dic6, S3×D4, S3×Q8, S3×C10, D4×C10, Q8×C10, C12⋊Q8, C5×Dic6, 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
(1 231 465 469 450)(2 232 466 470 451)(3 233 467 471 452)(4 234 468 472 453)(5 235 457 473 454)(6 236 458 474 455)(7 237 459 475 456)(8 238 460 476 445)(9 239 461 477 446)(10 240 462 478 447)(11 229 463 479 448)(12 230 464 480 449)(13 132 302 222 177)(14 121 303 223 178)(15 122 304 224 179)(16 123 305 225 180)(17 124 306 226 169)(18 125 307 227 170)(19 126 308 228 171)(20 127 309 217 172)(21 128 310 218 173)(22 129 311 219 174)(23 130 312 220 175)(24 131 301 221 176)(25 413 379 276 243)(26 414 380 265 244)(27 415 381 266 245)(28 416 382 267 246)(29 417 383 268 247)(30 418 384 269 248)(31 419 373 270 249)(32 420 374 271 250)(33 409 375 272 251)(34 410 376 273 252)(35 411 377 274 241)(36 412 378 275 242)(37 426 321 300 80)(38 427 322 289 81)(39 428 323 290 82)(40 429 324 291 83)(41 430 313 292 84)(42 431 314 293 73)(43 432 315 294 74)(44 421 316 295 75)(45 422 317 296 76)(46 423 318 297 77)(47 424 319 298 78)(48 425 320 299 79)(49 119 134 405 157)(50 120 135 406 158)(51 109 136 407 159)(52 110 137 408 160)(53 111 138 397 161)(54 112 139 398 162)(55 113 140 399 163)(56 114 141 400 164)(57 115 142 401 165)(58 116 143 402 166)(59 117 144 403 167)(60 118 133 404 168)(61 284 346 439 147)(62 285 347 440 148)(63 286 348 441 149)(64 287 337 442 150)(65 288 338 443 151)(66 277 339 444 152)(67 278 340 433 153)(68 279 341 434 154)(69 280 342 435 155)(70 281 343 436 156)(71 282 344 437 145)(72 283 345 438 146)(85 367 208 355 258)(86 368 209 356 259)(87 369 210 357 260)(88 370 211 358 261)(89 371 212 359 262)(90 372 213 360 263)(91 361 214 349 264)(92 362 215 350 253)(93 363 216 351 254)(94 364 205 352 255)(95 365 206 353 256)(96 366 207 354 257)(97 203 327 187 388)(98 204 328 188 389)(99 193 329 189 390)(100 194 330 190 391)(101 195 331 191 392)(102 196 332 192 393)(103 197 333 181 394)(104 198 334 182 395)(105 199 335 183 396)(106 200 336 184 385)(107 201 325 185 386)(108 202 326 186 387)
(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)
(1 396 438 373)(2 391 439 380)(3 386 440 375)(4 393 441 382)(5 388 442 377)(6 395 443 384)(7 390 444 379)(8 385 433 374)(9 392 434 381)(10 387 435 376)(11 394 436 383)(12 389 437 378)(13 324 213 164)(14 319 214 159)(15 314 215 166)(16 321 216 161)(17 316 205 168)(18 323 206 163)(19 318 207 158)(20 313 208 165)(21 320 209 160)(22 315 210 167)(23 322 211 162)(24 317 212 157)(25 475 329 277)(26 470 330 284)(27 477 331 279)(28 472 332 286)(29 479 333 281)(30 474 334 288)(31 469 335 283)(32 476 336 278)(33 471 325 285)(34 478 326 280)(35 473 327 287)(36 480 328 282)(37 93 138 225)(38 88 139 220)(39 95 140 227)(40 90 141 222)(41 85 142 217)(42 92 143 224)(43 87 144 219)(44 94 133 226)(45 89 134 221)(46 96 135 228)(47 91 136 223)(48 86 137 218)(49 131 296 359)(50 126 297 354)(51 121 298 349)(52 128 299 356)(53 123 300 351)(54 130 289 358)(55 125 290 353)(56 132 291 360)(57 127 292 355)(58 122 293 350)(59 129 294 357)(60 124 295 352)(61 244 466 194)(62 251 467 201)(63 246 468 196)(64 241 457 203)(65 248 458 198)(66 243 459 193)(67 250 460 200)(68 245 461 195)(69 252 462 202)(70 247 463 197)(71 242 464 204)(72 249 465 199)(73 253 116 304)(74 260 117 311)(75 255 118 306)(76 262 119 301)(77 257 120 308)(78 264 109 303)(79 259 110 310)(80 254 111 305)(81 261 112 312)(82 256 113 307)(83 263 114 302)(84 258 115 309)(97 150 274 235)(98 145 275 230)(99 152 276 237)(100 147 265 232)(101 154 266 239)(102 149 267 234)(103 156 268 229)(104 151 269 236)(105 146 270 231)(106 153 271 238)(107 148 272 233)(108 155 273 240)(169 421 364 404)(170 428 365 399)(171 423 366 406)(172 430 367 401)(173 425 368 408)(174 432 369 403)(175 427 370 398)(176 422 371 405)(177 429 372 400)(178 424 361 407)(179 431 362 402)(180 426 363 397)(181 343 417 448)(182 338 418 455)(183 345 419 450)(184 340 420 445)(185 347 409 452)(186 342 410 447)(187 337 411 454)(188 344 412 449)(189 339 413 456)(190 346 414 451)(191 341 415 446)(192 348 416 453)
(1 303 438 264)(2 308 439 257)(3 301 440 262)(4 306 441 255)(5 311 442 260)(6 304 443 253)(7 309 444 258)(8 302 433 263)(9 307 434 256)(10 312 435 261)(11 305 436 254)(12 310 437 259)(13 278 213 476)(14 283 214 469)(15 288 215 474)(16 281 216 479)(17 286 205 472)(18 279 206 477)(19 284 207 470)(20 277 208 475)(21 282 209 480)(22 287 210 473)(23 280 211 478)(24 285 212 471)(25 313 329 165)(26 318 330 158)(27 323 331 163)(28 316 332 168)(29 321 333 161)(30 314 334 166)(31 319 335 159)(32 324 336 164)(33 317 325 157)(34 322 326 162)(35 315 327 167)(36 320 328 160)(37 103 138 268)(38 108 139 273)(39 101 140 266)(40 106 141 271)(41 99 142 276)(42 104 143 269)(43 97 144 274)(44 102 133 267)(45 107 134 272)(46 100 135 265)(47 105 136 270)(48 98 137 275)(49 409 296 185)(50 414 297 190)(51 419 298 183)(52 412 299 188)(53 417 300 181)(54 410 289 186)(55 415 290 191)(56 420 291 184)(57 413 292 189)(58 418 293 182)(59 411 294 187)(60 416 295 192)(61 366 466 171)(62 371 467 176)(63 364 468 169)(64 369 457 174)(65 362 458 179)(66 367 459 172)(67 372 460 177)(68 365 461 170)(69 370 462 175)(70 363 463 180)(71 368 464 173)(72 361 465 178)(73 395 116 384)(74 388 117 377)(75 393 118 382)(76 386 119 375)(77 391 120 380)(78 396 109 373)(79 389 110 378)(80 394 111 383)(81 387 112 376)(82 392 113 381)(83 385 114 374)(84 390 115 379)(85 237 217 152)(86 230 218 145)(87 235 219 150)(88 240 220 155)(89 233 221 148)(90 238 222 153)(91 231 223 146)(92 236 224 151)(93 229 225 156)(94 234 226 149)(95 239 227 154)(96 232 228 147)(121 345 349 450)(122 338 350 455)(123 343 351 448)(124 348 352 453)(125 341 353 446)(126 346 354 451)(127 339 355 456)(128 344 356 449)(129 337 357 454)(130 342 358 447)(131 347 359 452)(132 340 360 445)(193 401 243 430)(194 406 244 423)(195 399 245 428)(196 404 246 421)(197 397 247 426)(198 402 248 431)(199 407 249 424)(200 400 250 429)(201 405 251 422)(202 398 252 427)(203 403 241 432)(204 408 242 425)

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

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

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

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

׿
×
𝔽