Copied to
clipboard

## G = Dic5×C2×C12order 480 = 25·3·5

### Direct product of C2×C12 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — Dic5×C2×C12
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C2×C6×Dic5 — Dic5×C2×C12
 Lower central C5 — Dic5×C2×C12
 Upper central C1 — C22×C12

Generators and relations for Dic5×C2×C12
G = < a,b,c,d | a2=b12=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 432 in 216 conjugacy classes, 162 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C12, C2×C12, C22×C6, C30, C30, C2×C42, C2×Dic5, C2×C20, C22×C10, C4×C12, C22×C12, C22×C12, C3×Dic5, C60, C2×C30, C2×C30, C4×Dic5, C22×Dic5, C22×C20, C2×C4×C12, C6×Dic5, C2×C60, C22×C30, C2×C4×Dic5, C12×Dic5, C2×C6×Dic5, C22×C60, Dic5×C2×C12
Quotients:

Smallest permutation representation of Dic5×C2×C12
Regular action on 480 points
Generators in S480
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 347)(14 348)(15 337)(16 338)(17 339)(18 340)(19 341)(20 342)(21 343)(22 344)(23 345)(24 346)(25 371)(26 372)(27 361)(28 362)(29 363)(30 364)(31 365)(32 366)(33 367)(34 368)(35 369)(36 370)(49 87)(50 88)(51 89)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 85)(60 86)(61 377)(62 378)(63 379)(64 380)(65 381)(66 382)(67 383)(68 384)(69 373)(70 374)(71 375)(72 376)(73 427)(74 428)(75 429)(76 430)(77 431)(78 432)(79 421)(80 422)(81 423)(82 424)(83 425)(84 426)(97 239)(98 240)(99 229)(100 230)(101 231)(102 232)(103 233)(104 234)(105 235)(106 236)(107 237)(108 238)(109 170)(110 171)(111 172)(112 173)(113 174)(114 175)(115 176)(116 177)(117 178)(118 179)(119 180)(120 169)(121 207)(122 208)(123 209)(124 210)(125 211)(126 212)(127 213)(128 214)(129 215)(130 216)(131 205)(132 206)(133 441)(134 442)(135 443)(136 444)(137 433)(138 434)(139 435)(140 436)(141 437)(142 438)(143 439)(144 440)(145 475)(146 476)(147 477)(148 478)(149 479)(150 480)(151 469)(152 470)(153 471)(154 472)(155 473)(156 474)(157 266)(158 267)(159 268)(160 269)(161 270)(162 271)(163 272)(164 273)(165 274)(166 275)(167 276)(168 265)(181 457)(182 458)(183 459)(184 460)(185 461)(186 462)(187 463)(188 464)(189 465)(190 466)(191 467)(192 468)(193 290)(194 291)(195 292)(196 293)(197 294)(198 295)(199 296)(200 297)(201 298)(202 299)(203 300)(204 289)(217 355)(218 356)(219 357)(220 358)(221 359)(222 360)(223 349)(224 350)(225 351)(226 352)(227 353)(228 354)(241 332)(242 333)(243 334)(244 335)(245 336)(246 325)(247 326)(248 327)(249 328)(250 329)(251 330)(252 331)(253 283)(254 284)(255 285)(256 286)(257 287)(258 288)(259 277)(260 278)(261 279)(262 280)(263 281)(264 282)(301 448)(302 449)(303 450)(304 451)(305 452)(306 453)(307 454)(308 455)(309 456)(310 445)(311 446)(312 447)(313 419)(314 420)(315 409)(316 410)(317 411)(318 412)(319 413)(320 414)(321 415)(322 416)(323 417)(324 418)(385 404)(386 405)(387 406)(388 407)(389 408)(390 397)(391 398)(392 399)(393 400)(394 401)(395 402)(396 403)
(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)
(1 463 54 363 118 437 80 453 328 240)(2 464 55 364 119 438 81 454 329 229)(3 465 56 365 120 439 82 455 330 230)(4 466 57 366 109 440 83 456 331 231)(5 467 58 367 110 441 84 445 332 232)(6 468 59 368 111 442 73 446 333 233)(7 457 60 369 112 443 74 447 334 234)(8 458 49 370 113 444 75 448 335 235)(9 459 50 371 114 433 76 449 336 236)(10 460 51 372 115 434 77 450 325 237)(11 461 52 361 116 435 78 451 326 238)(12 462 53 362 117 436 79 452 327 239)(13 132 63 296 358 399 260 477 159 324)(14 121 64 297 359 400 261 478 160 313)(15 122 65 298 360 401 262 479 161 314)(16 123 66 299 349 402 263 480 162 315)(17 124 67 300 350 403 264 469 163 316)(18 125 68 289 351 404 253 470 164 317)(19 126 69 290 352 405 254 471 165 318)(20 127 70 291 353 406 255 472 166 319)(21 128 71 292 354 407 256 473 167 320)(22 129 72 293 355 408 257 474 168 321)(23 130 61 294 356 397 258 475 157 322)(24 131 62 295 357 398 259 476 158 323)(25 175 137 430 302 245 106 39 183 88)(26 176 138 431 303 246 107 40 184 89)(27 177 139 432 304 247 108 41 185 90)(28 178 140 421 305 248 97 42 186 91)(29 179 141 422 306 249 98 43 187 92)(30 180 142 423 307 250 99 44 188 93)(31 169 143 424 308 251 100 45 189 94)(32 170 144 425 309 252 101 46 190 95)(33 171 133 426 310 241 102 47 191 96)(34 172 134 427 311 242 103 48 192 85)(35 173 135 428 312 243 104 37 181 86)(36 174 136 429 301 244 105 38 182 87)(145 266 416 345 216 377 197 218 390 288)(146 267 417 346 205 378 198 219 391 277)(147 268 418 347 206 379 199 220 392 278)(148 269 419 348 207 380 200 221 393 279)(149 270 420 337 208 381 201 222 394 280)(150 271 409 338 209 382 202 223 395 281)(151 272 410 339 210 383 203 224 396 282)(152 273 411 340 211 384 204 225 385 283)(153 274 412 341 212 373 193 226 386 284)(154 275 413 342 213 374 194 227 387 285)(155 276 414 343 214 375 195 228 388 286)(156 265 415 344 215 376 196 217 389 287)
(1 18 437 404)(2 19 438 405)(3 20 439 406)(4 21 440 407)(5 22 441 408)(6 23 442 397)(7 24 443 398)(8 13 444 399)(9 14 433 400)(10 15 434 401)(11 16 435 402)(12 17 436 403)(25 148 245 380)(26 149 246 381)(27 150 247 382)(28 151 248 383)(29 152 249 384)(30 153 250 373)(31 154 251 374)(32 155 252 375)(33 156 241 376)(34 145 242 377)(35 146 243 378)(36 147 244 379)(37 346 135 391)(38 347 136 392)(39 348 137 393)(40 337 138 394)(41 338 139 395)(42 339 140 396)(43 340 141 385)(44 341 142 386)(45 342 143 387)(46 343 144 388)(47 344 133 389)(48 345 134 390)(49 159 448 296)(50 160 449 297)(51 161 450 298)(52 162 451 299)(53 163 452 300)(54 164 453 289)(55 165 454 290)(56 166 455 291)(57 167 456 292)(58 168 445 293)(59 157 446 294)(60 158 447 295)(61 368 475 333)(62 369 476 334)(63 370 477 335)(64 371 478 336)(65 372 479 325)(66 361 480 326)(67 362 469 327)(68 363 470 328)(69 364 471 329)(70 365 472 330)(71 366 473 331)(72 367 474 332)(73 356 468 322)(74 357 457 323)(75 358 458 324)(76 359 459 313)(77 360 460 314)(78 349 461 315)(79 350 462 316)(80 351 463 317)(81 352 464 318)(82 353 465 319)(83 354 466 320)(84 355 467 321)(85 266 311 197)(86 267 312 198)(87 268 301 199)(88 269 302 200)(89 270 303 201)(90 271 304 202)(91 272 305 203)(92 273 306 204)(93 274 307 193)(94 275 308 194)(95 276 309 195)(96 265 310 196)(97 210 178 282)(98 211 179 283)(99 212 180 284)(100 213 169 285)(101 214 170 286)(102 215 171 287)(103 216 172 288)(104 205 173 277)(105 206 174 278)(106 207 175 279)(107 208 176 280)(108 209 177 281)(109 256 231 128)(110 257 232 129)(111 258 233 130)(112 259 234 131)(113 260 235 132)(114 261 236 121)(115 262 237 122)(116 263 238 123)(117 264 239 124)(118 253 240 125)(119 254 229 126)(120 255 230 127)(181 417 428 219)(182 418 429 220)(183 419 430 221)(184 420 431 222)(185 409 432 223)(186 410 421 224)(187 411 422 225)(188 412 423 226)(189 413 424 227)(190 414 425 228)(191 415 426 217)(192 416 427 218)

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

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

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

192 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 4I ··· 4X 5A 5B 6A ··· 6N 10A ··· 10N 12A ··· 12P 12Q ··· 12AV 15A 15B 15C 15D 20A ··· 20P 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 3 3 4 ··· 4 4 ··· 4 5 5 6 ··· 6 10 ··· 10 12 ··· 12 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 1 1 1 ··· 1 5 ··· 5 2 2 1 ··· 1 2 ··· 2 1 ··· 1 5 ··· 5 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

192 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 type + + + + + - + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 D5 Dic5 D10 D10 C3×D5 C4×D5 C3×Dic5 C6×D5 C6×D5 D5×C12 kernel Dic5×C2×C12 C12×Dic5 C2×C6×Dic5 C22×C60 C2×C4×Dic5 C6×Dic5 C2×C60 C4×Dic5 C22×Dic5 C22×C20 C2×Dic5 C2×C20 C22×C12 C2×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 2 16 8 8 4 2 32 16 2 8 4 2 4 16 16 8 4 32

Matrix representation of Dic5×C2×C12 in GL4(𝔽61) generated by

 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 1
,
 48 0 0 0 0 50 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 60 0 0 0 0 1 60 0 0 19 43
,
 60 0 0 0 0 50 0 0 0 0 52 29 0 0 14 9
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1],[48,0,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,60,0,0,0,0,1,19,0,0,60,43],[60,0,0,0,0,50,0,0,0,0,52,14,0,0,29,9] >;

Dic5×C2×C12 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_2\times C_{12}
% in TeX

G:=Group("Dic5xC2xC12");
// GroupNames label

G:=SmallGroup(480,715);
// by ID

G=gap.SmallGroup(480,715);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,268,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽