Copied to
clipboard

## G = C2×C5⋊Dic12order 480 = 25·3·5

### Direct product of C2 and C5⋊Dic12

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 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C10, C10 [×2], Dic3 [×4], C12 [×2], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C24 [×2], Dic6 [×2], Dic6 [×4], C2×Dic3 [×2], C2×C12, C30, C30 [×2], C2×Q16, C52C8 [×2], Dic10 [×3], C2×Dic5, C2×C20, C2×C20, C5×Q8 [×3], Dic12 [×4], C2×C24, C2×Dic6, C2×Dic6, C5×Dic3 [×2], Dic15 [×2], C60 [×2], C2×C30, C2×C52C8, C5⋊Q16 [×4], C2×Dic10, Q8×C10, C2×Dic12, C3×C52C8 [×2], C5×Dic6 [×2], C5×Dic6, C10×Dic3, Dic30 [×2], Dic30, C2×Dic15, C2×C60, C2×C5⋊Q16, C5⋊Dic12 [×4], C6×C52C8, C10×Dic6, C2×Dic30, C2×C5⋊Dic12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], Q16 [×2], C2×D4, D10 [×3], D12 [×2], C22×S3, C2×Q16, C5⋊D4 [×2], C22×D5, Dic12 [×2], C2×D12, S3×D5, C5⋊Q16 [×2], C2×C5⋊D4, C2×Dic12, C5⋊D12 [×2], C2×S3×D5, C2×C5⋊Q16, C5⋊Dic12 [×2], C2×C5⋊D12, C2×C5⋊Dic12

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

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 15A 15B 20A 20B 20C 20D 20E ··· 20L 24A ··· 24H 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 15 15 20 20 20 20 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 12 12 60 60 2 2 2 2 2 10 10 10 10 2 ··· 2 2 2 2 2 4 4 4 4 4 4 12 ··· 12 10 ··· 10 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + - + + + + - + - + + + - image C1 C2 C2 C2 C2 S3 D4 D4 D5 D6 D6 Q16 D10 D10 D12 D12 C5⋊D4 C5⋊D4 Dic12 S3×D5 C5⋊Q16 C5⋊D12 C2×S3×D5 C5⋊D12 C5⋊Dic12 kernel C2×C5⋊Dic12 C5⋊Dic12 C6×C5⋊2C8 C10×Dic6 C2×Dic30 C2×C5⋊2C8 C60 C2×C30 C2×Dic6 C5⋊2C8 C2×C20 C30 Dic6 C2×C12 C20 C2×C10 C12 C2×C6 C10 C2×C4 C6 C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 1 2 2 1 4 4 2 2 2 4 4 8 2 4 2 2 2 8

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

 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
,
 51 240 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 1 0 0 0 0 0 0 1
,
 0 240 0 0 0 0 240 0 0 0 0 0 0 0 219 57 0 0 0 0 93 0 0 0 0 0 0 0 0 1 0 0 0 0 240 1
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 158 143 0 0 0 0 149 83 0 0 0 0 0 0 225 139 0 0 0 0 123 16

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

׿
×
𝔽