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## G = C3×C4.Dic10order 480 = 25·3·5

### Direct product of C3 and C4.Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C3×C4.Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — C6×Dic5 — C12×Dic5 — C3×C4.Dic10
 Lower central C5 — C2×C10 — C3×C4.Dic10
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×C4.Dic10
G = < a,b,c,d | a3=b4=c20=1, d2=c10, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b2c-1 >

Subgroups: 288 in 112 conjugacy classes, 66 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, C2×C4, C10, C12, C12, C2×C6, C15, C42, C4⋊C4, C4⋊C4, Dic5, C20, C20, C2×C10, C2×C12, C2×C12, C2×C12, C30, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C3×Dic5, C60, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C3×C42.C2, C6×Dic5, C6×Dic5, C2×C60, C2×C60, C4.Dic10, C12×Dic5, C3×C10.D4, C3×C4⋊Dic5, C3×C4⋊Dic5, C15×C4⋊C4, C3×C4.Dic10
Quotients: C1, C2, C3, C22, C6, Q8, C23, D5, C2×C6, C2×Q8, C4○D4, D10, C3×Q8, C22×C6, C3×D5, C42.C2, Dic10, C22×D5, C6×Q8, C3×C4○D4, C6×D5, C2×Dic10, D42D5, Q82D5, C3×C42.C2, C3×Dic10, D5×C2×C6, C4.Dic10, C6×Dic10, C3×D42D5, C3×Q82D5, C3×C4.Dic10

Smallest permutation representation of C3×C4.Dic10
Regular action on 480 points
Generators in S480
(1 63 190)(2 64 191)(3 65 192)(4 66 193)(5 67 194)(6 68 195)(7 69 196)(8 70 197)(9 71 198)(10 72 199)(11 73 200)(12 74 181)(13 75 182)(14 76 183)(15 77 184)(16 78 185)(17 79 186)(18 80 187)(19 61 188)(20 62 189)(21 143 422)(22 144 423)(23 145 424)(24 146 425)(25 147 426)(26 148 427)(27 149 428)(28 150 429)(29 151 430)(30 152 431)(31 153 432)(32 154 433)(33 155 434)(34 156 435)(35 157 436)(36 158 437)(37 159 438)(38 160 439)(39 141 440)(40 142 421)(41 352 329)(42 353 330)(43 354 331)(44 355 332)(45 356 333)(46 357 334)(47 358 335)(48 359 336)(49 360 337)(50 341 338)(51 342 339)(52 343 340)(53 344 321)(54 345 322)(55 346 323)(56 347 324)(57 348 325)(58 349 326)(59 350 327)(60 351 328)(81 472 454)(82 473 455)(83 474 456)(84 475 457)(85 476 458)(86 477 459)(87 478 460)(88 479 441)(89 480 442)(90 461 443)(91 462 444)(92 463 445)(93 464 446)(94 465 447)(95 466 448)(96 467 449)(97 468 450)(98 469 451)(99 470 452)(100 471 453)(101 365 264)(102 366 265)(103 367 266)(104 368 267)(105 369 268)(106 370 269)(107 371 270)(108 372 271)(109 373 272)(110 374 273)(111 375 274)(112 376 275)(113 377 276)(114 378 277)(115 379 278)(116 380 279)(117 361 280)(118 362 261)(119 363 262)(120 364 263)(121 392 418)(122 393 419)(123 394 420)(124 395 401)(125 396 402)(126 397 403)(127 398 404)(128 399 405)(129 400 406)(130 381 407)(131 382 408)(132 383 409)(133 384 410)(134 385 411)(135 386 412)(136 387 413)(137 388 414)(138 389 415)(139 390 416)(140 391 417)(161 241 228)(162 242 229)(163 243 230)(164 244 231)(165 245 232)(166 246 233)(167 247 234)(168 248 235)(169 249 236)(170 250 237)(171 251 238)(172 252 239)(173 253 240)(174 254 221)(175 255 222)(176 256 223)(177 257 224)(178 258 225)(179 259 226)(180 260 227)(201 307 284)(202 308 285)(203 309 286)(204 310 287)(205 311 288)(206 312 289)(207 313 290)(208 314 291)(209 315 292)(210 316 293)(211 317 294)(212 318 295)(213 319 296)(214 320 297)(215 301 298)(216 302 299)(217 303 300)(218 304 281)(219 305 282)(220 306 283)
(1 222 275 33)(2 34 276 223)(3 224 277 35)(4 36 278 225)(5 226 279 37)(6 38 280 227)(7 228 261 39)(8 40 262 229)(9 230 263 21)(10 22 264 231)(11 232 265 23)(12 24 266 233)(13 234 267 25)(14 26 268 235)(15 236 269 27)(16 28 270 237)(17 238 271 29)(18 30 272 239)(19 240 273 31)(20 32 274 221)(41 290 396 87)(42 88 397 291)(43 292 398 89)(44 90 399 293)(45 294 400 91)(46 92 381 295)(47 296 382 93)(48 94 383 297)(49 298 384 95)(50 96 385 299)(51 300 386 97)(52 98 387 281)(53 282 388 99)(54 100 389 283)(55 284 390 81)(56 82 391 285)(57 286 392 83)(58 84 393 287)(59 288 394 85)(60 86 395 289)(61 173 110 153)(62 154 111 174)(63 175 112 155)(64 156 113 176)(65 177 114 157)(66 158 115 178)(67 179 116 159)(68 160 117 180)(69 161 118 141)(70 142 119 162)(71 163 120 143)(72 144 101 164)(73 165 102 145)(74 146 103 166)(75 167 104 147)(76 148 105 168)(77 169 106 149)(78 150 107 170)(79 171 108 151)(80 152 109 172)(121 456 325 309)(122 310 326 457)(123 458 327 311)(124 312 328 459)(125 460 329 313)(126 314 330 441)(127 442 331 315)(128 316 332 443)(129 444 333 317)(130 318 334 445)(131 446 335 319)(132 320 336 447)(133 448 337 301)(134 302 338 449)(135 450 339 303)(136 304 340 451)(137 452 321 305)(138 306 322 453)(139 454 323 307)(140 308 324 455)(181 425 367 246)(182 247 368 426)(183 427 369 248)(184 249 370 428)(185 429 371 250)(186 251 372 430)(187 431 373 252)(188 253 374 432)(189 433 375 254)(190 255 376 434)(191 435 377 256)(192 257 378 436)(193 437 379 258)(194 259 380 438)(195 439 361 260)(196 241 362 440)(197 421 363 242)(198 243 364 422)(199 423 365 244)(200 245 366 424)(201 416 472 346)(202 347 473 417)(203 418 474 348)(204 349 475 419)(205 420 476 350)(206 351 477 401)(207 402 478 352)(208 353 479 403)(209 404 480 354)(210 355 461 405)(211 406 462 356)(212 357 463 407)(213 408 464 358)(214 359 465 409)(215 410 466 360)(216 341 467 411)(217 412 468 342)(218 343 469 413)(219 414 470 344)(220 345 471 415)
(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)
(1 475 11 465)(2 203 12 213)(3 473 13 463)(4 201 14 211)(5 471 15 461)(6 219 16 209)(7 469 17 479)(8 217 18 207)(9 467 19 477)(10 215 20 205)(21 341 31 351)(22 410 32 420)(23 359 33 349)(24 408 34 418)(25 357 35 347)(26 406 36 416)(27 355 37 345)(28 404 38 414)(29 353 39 343)(30 402 40 412)(41 242 51 252)(42 440 52 430)(43 260 53 250)(44 438 54 428)(45 258 55 248)(46 436 56 426)(47 256 57 246)(48 434 58 424)(49 254 59 244)(50 432 60 422)(61 459 71 449)(62 311 72 301)(63 457 73 447)(64 309 74 319)(65 455 75 445)(66 307 76 317)(67 453 77 443)(68 305 78 315)(69 451 79 441)(70 303 80 313)(81 369 91 379)(82 182 92 192)(83 367 93 377)(84 200 94 190)(85 365 95 375)(86 198 96 188)(87 363 97 373)(88 196 98 186)(89 361 99 371)(90 194 100 184)(101 448 111 458)(102 320 112 310)(103 446 113 456)(104 318 114 308)(105 444 115 454)(106 316 116 306)(107 442 117 452)(108 314 118 304)(109 460 119 450)(110 312 120 302)(121 146 131 156)(122 165 132 175)(123 144 133 154)(124 163 134 173)(125 142 135 152)(126 161 136 171)(127 160 137 150)(128 179 138 169)(129 158 139 148)(130 177 140 167)(141 340 151 330)(143 338 153 328)(145 336 155 326)(147 334 157 324)(149 332 159 322)(162 339 172 329)(164 337 174 327)(166 335 176 325)(168 333 178 323)(170 331 180 321)(181 296 191 286)(183 294 193 284)(185 292 195 282)(187 290 197 300)(189 288 199 298)(202 267 212 277)(204 265 214 275)(206 263 216 273)(208 261 218 271)(210 279 220 269)(221 350 231 360)(222 419 232 409)(223 348 233 358)(224 417 234 407)(225 346 235 356)(226 415 236 405)(227 344 237 354)(228 413 238 403)(229 342 239 352)(230 411 240 401)(241 387 251 397)(243 385 253 395)(245 383 255 393)(247 381 257 391)(249 399 259 389)(262 468 272 478)(264 466 274 476)(266 464 276 474)(268 462 278 472)(270 480 280 470)(281 372 291 362)(283 370 293 380)(285 368 295 378)(287 366 297 376)(289 364 299 374)(382 435 392 425)(384 433 394 423)(386 431 396 421)(388 429 398 439)(390 427 400 437)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A ··· 6F 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12P 12Q 12R 12S 12T 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X order 1 2 2 2 3 3 4 4 4 4 4 4 4 4 4 4 5 5 6 ··· 6 10 ··· 10 12 12 12 12 12 12 12 12 12 ··· 12 12 12 12 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 1 1 2 2 4 4 10 10 10 10 20 20 2 2 1 ··· 1 2 ··· 2 2 2 2 2 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + + - - + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 Q8 D5 C4○D4 D10 C3×Q8 C3×D5 Dic10 C3×C4○D4 C6×D5 C3×Dic10 D4⋊2D5 Q8⋊2D5 C3×D4⋊2D5 C3×Q8⋊2D5 kernel C3×C4.Dic10 C12×Dic5 C3×C10.D4 C3×C4⋊Dic5 C15×C4⋊C4 C4.Dic10 C4×Dic5 C10.D4 C4⋊Dic5 C5×C4⋊C4 C60 C3×C4⋊C4 C30 C2×C12 C20 C4⋊C4 C12 C10 C2×C4 C4 C6 C6 C2 C2 # reps 1 1 2 3 1 2 2 4 6 2 2 2 4 6 4 4 8 8 12 16 2 2 4 4

Matrix representation of C3×C4.Dic10 in GL4(𝔽61) generated by

 47 0 0 0 0 47 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 0 60 0 0 0 0 11 0 0 0 54 50
,
 25 32 0 0 2 27 0 0 0 0 21 5 0 0 34 40
,
 5 56 0 0 54 56 0 0 0 0 1 0 0 0 16 60
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,11,54,0,0,0,50],[25,2,0,0,32,27,0,0,0,0,21,34,0,0,5,40],[5,54,0,0,56,56,0,0,0,0,1,16,0,0,0,60] >;

C3×C4.Dic10 in GAP, Magma, Sage, TeX

C_3\times C_4.{\rm Dic}_{10}
% in TeX

G:=Group("C3xC4.Dic10");
// GroupNames label

G:=SmallGroup(480,683);
// by ID

G=gap.SmallGroup(480,683);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,344,1598,555,142,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=c^20=1,d^2=c^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations

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