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G = C2×C5⋊Dic12  order 480 = 25·3·5

Direct product of C2 and C5⋊Dic12

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

Aliases: C2×C5⋊Dic12, C30⋊3Q16, C60.66D4, C10⋊2Dic12, C20.17D12, C60.132C23, Dic6.33D10, Dic30.52C22, C15⋊6(C2×Q16), C5⋊3(C2×Dic12), C30.95(C2×D4), C6⋊1(C5⋊Q16), (C2×C30).63D4, C5⋊2C8.36D6, C10.53(C2×D12), (C2×C10).42D12, (C2×C20).102D6, C4.8(C5⋊D12), (C2×Dic6).2D5, (C2×C12).292D10, C12.58(C5⋊D4), C20.99(C22×S3), (C10×Dic6).3C2, (C2×C60).136C22, (C2×Dic30).17C2, C12.155(C22×D5), C22.22(C5⋊D12), (C5×Dic6).38C22, C4.80(C2×S3×D5), C3⋊1(C2×C5⋊Q16), C6.7(C2×C5⋊D4), (C6×C5⋊2C8).6C2, (C2×C5⋊2C8).6S3, (C2×C4).149(S3×D5), C2.11(C2×C5⋊D12), (C2×C6).34(C5⋊D4), (C3×C5⋊2C8).40C22, SmallGroup(480,396)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C60 — C2×C5⋊Dic12
Chief series
C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — C5⋊Dic12 — C2×C5⋊Dic12
Lower central
C15 — C30 — C60 — C2×C5⋊Dic12
Upper central
C1 — C22 — C2×C4

Generators and relations for C2×C5⋊Dic12
 G = < a,b,c,d | a2=b5=c24=1, d2=c12, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 540 in 120 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C2×C6, C15, C2×C8, Q16, C2×Q8, Dic5, C20, C20, C2×C10, C24, Dic6, Dic6, C2×Dic3, C2×C12, C30, C30, C2×Q16, C5⋊2C8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, Dic12, C2×C24, C2×Dic6, C2×Dic6, C5×Dic3, Dic15, C60, C2×C30, C2×C5⋊2C8, C5⋊Q16, C2×Dic10, Q8×C10, C2×Dic12, C3×C5⋊2C8, C5×Dic6, C5×Dic6, C10×Dic3, Dic30, Dic30, C2×Dic15, C2×C60, C2×C5⋊Q16, C5⋊Dic12, C6×C5⋊2C8, C10×Dic6, C2×Dic30, C2×C5⋊Dic12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, Q16, C2×D4, D10, D12, C22×S3, C2×Q16, C5⋊D4, C22×D5, Dic12, C2×D12, S3×D5, C5⋊Q16, C2×C5⋊D4, C2×Dic12, C5⋊D12, C2×S3×D5, C2×C5⋊Q16, C5⋊Dic12, C2×C5⋊D12, C2×C5⋊Dic12

Smallest permutation representation of C2×C5⋊Dic12
►Regular action on 480 points
Generators in S480
(1 47)(2 48)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 31)(10 32)(11 33)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(49 223)(50 224)(51 225)(52 226)(53 227)(54 228)(55 229)(56 230)(57 231)(58 232)(59 233)(60 234)(61 235)(62 236)(63 237)(64 238)(65 239)(66 240)(67 217)(68 218)(69 219)(70 220)(71 221)(72 222)(73 265)(74 266)(75 267)(76 268)(77 269)(78 270)(79 271)(80 272)(81 273)(82 274)(83 275)(84 276)(85 277)(86 278)(87 279)(88 280)(89 281)(90 282)(91 283)(92 284)(93 285)(94 286)(95 287)(96 288)(97 454)(98 455)(99 456)(100 433)(101 434)(102 435)(103 436)(104 437)(105 438)(106 439)(107 440)(108 441)(109 442)(110 443)(111 444)(112 445)(113 446)(114 447)(115 448)(116 449)(117 450)(118 451)(119 452)(120 453)(121 466)(122 467)(123 468)(124 469)(125 470)(126 471)(127 472)(128 473)(129 474)(130 475)(131 476)(132 477)(133 478)(134 479)(135 480)(136 457)(137 458)(138 459)(139 460)(140 461)(141 462)(142 463)(143 464)(144 465)(145 195)(146 196)(147 197)(148 198)(149 199)(150 200)(151 201)(152 202)(153 203)(154 204)(155 205)(156 206)(157 207)(158 208)(159 209)(160 210)(161 211)(162 212)(163 213)(164 214)(165 215)(166 216)(167 193)(168 194)(169 245)(170 246)(171 247)(172 248)(173 249)(174 250)(175 251)(176 252)(177 253)(178 254)(179 255)(180 256)(181 257)(182 258)(183 259)(184 260)(185 261)(186 262)(187 263)(188 264)(189 241)(190 242)(191 243)(192 244)(289 430)(290 431)(291 432)(292 409)(293 410)(294 411)(295 412)(296 413)(297 414)(298 415)(299 416)(300 417)(301 418)(302 419)(303 420)(304 421)(305 422)(306 423)(307 424)(308 425)(309 426)(310 427)(311 428)(312 429)(313 365)(314 366)(315 367)(316 368)(317 369)(318 370)(319 371)(320 372)(321 373)(322 374)(323 375)(324 376)(325 377)(326 378)(327 379)(328 380)(329 381)(330 382)(331 383)(332 384)(333 361)(334 362)(335 363)(336 364)(337 403)(338 404)(339 405)(340 406)(341 407)(342 408)(343 385)(344 386)(345 387)(346 388)(347 389)(348 390)(349 391)(350 392)(351 393)(352 394)(353 395)(354 396)(355 397)(356 398)(357 399)(358 400)(359 401)(360 402)

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

(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)

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

66 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L24A···24H30A···30F60A···60H
order1222344444455666888810···101212121215152020202020···2024···2430···3060···60
size11112221212606022222101010102···2222244444412···1210···104···44···4

66 irreducible representations

dim1111122222222222222444444
type+++++++++++-++++-+-+++-
imageC1C2C2C2C2S3D4D4D5D6D6Q16D10D10D12D12C5⋊D4C5⋊D4Dic12S3×D5C5⋊Q16C5⋊D12C2×S3×D5C5⋊D12C5⋊Dic12
kernelC2×C5⋊Dic12C5⋊Dic12C6×C5⋊2C8C10×Dic6C2×Dic30C2×C5⋊2C8C60C2×C30C2×Dic6C5⋊2C8C2×C20C30Dic6C2×C12C20C2×C10C12C2×C6C10C2×C4C6C4C4C22C2
# reps1411111122144222448242228

Matrix representation of C2×C5⋊Dic12 ►in GL6(𝔽241)

24000000
02400000
00240000
00024000
000010
000001
,
512400000
100000
001000
000100
000010
000001
,
02400000
24000000
002195700
0093000
000001
00002401
,
24000000
02400000
0015814300
001498300
0000225139
000012316

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,1,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,240,0,0,0,0,240,0,0,0,0,0,0,0,219,93,0,0,0,0,57,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,158,149,0,0,0,0,143,83,0,0,0,0,0,0,225,123,0,0,0,0,139,16] >;
 

C2×C5⋊Dic12 in GAP, Magma, Sage, TeX 

C_2\times C_5\rtimes {\rm Dic}_{12}
 
% in TeX
 
G:=Group("C2xC5:Dic12");
 
// GroupNames label
 
G:=SmallGroup(480,396);
 
// by ID
 
G=gap.SmallGroup(480,396);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,120,346,80,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^5=c^24=1,d^2=c^12,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=c^-1>;
 
// generators/relations
 

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