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G = C3×C4.Dic10order 480 = 25·3·5

Direct product of C3 and C4.Dic10

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

Aliases: C3×C4.Dic10, C60.18Q8, C12.18Dic10, C20.3(C3×Q8), C10.5(C6×Q8), C4⋊Dic5.7C6, C30.78(C2×Q8), (C4×Dic5).2C6, C2.8(C6×Dic10), C4.3(C3×Dic10), (C2×C12).233D10, C6.46(C2×Dic10), C1519(C42.C2), C10.D4.3C6, C30.234(C4○D4), (C2×C60).284C22, (C2×C30).348C23, C6.47(Q82D5), (C12×Dic5).12C2, C6.115(D42D5), (C6×Dic5).240C22, (C5×C4⋊C4).7C6, C4⋊C4.6(C3×D5), C53(C3×C42.C2), (C2×C20).5(C2×C6), (C3×C4⋊C4).13D5, (C2×C4).38(C6×D5), (C15×C4⋊C4).14C2, C22.48(D5×C2×C6), C10.32(C3×C4○D4), C2.4(C3×Q82D5), C2.12(C3×D42D5), (C3×C4⋊Dic5).21C2, (C2×C10).31(C22×C6), (C2×Dic5).30(C2×C6), (C3×C10.D4).8C2, (C2×C6).344(C22×D5), SmallGroup(480,683)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×C4.Dic10
C1C5C10C2×C10C2×C30C6×Dic5C12×Dic5 — C3×C4.Dic10
C5C2×C10 — C3×C4.Dic10
C1C2×C6C3×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 [×3], C3, C4 [×2], C4 [×6], C22, C5, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], C10 [×3], C12 [×2], C12 [×6], C2×C6, C15, C42, C4⋊C4, C4⋊C4 [×5], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C12, C2×C12 [×2], C2×C12 [×4], C30 [×3], C42.C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×C12, C3×C4⋊C4, C3×C4⋊C4 [×5], C3×Dic5 [×4], C60 [×2], C60 [×2], C2×C30, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C4⋊Dic5 [×2], C5×C4⋊C4, C3×C42.C2, C6×Dic5 [×2], C6×Dic5 [×2], C2×C60, C2×C60 [×2], C4.Dic10, C12×Dic5, C3×C10.D4 [×2], C3×C4⋊Dic5, C3×C4⋊Dic5 [×2], C15×C4⋊C4, C3×C4.Dic10
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], Q8 [×2], C23, D5, C2×C6 [×7], C2×Q8, C4○D4 [×2], D10 [×3], C3×Q8 [×2], C22×C6, C3×D5, C42.C2, Dic10 [×2], C22×D5, C6×Q8, C3×C4○D4 [×2], C6×D5 [×3], C2×Dic10, D42D5, Q82D5, C3×C42.C2, C3×Dic10 [×2], 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
(1 355 254)(2 356 255)(3 357 256)(4 358 257)(5 359 258)(6 360 259)(7 341 260)(8 342 241)(9 343 242)(10 344 243)(11 345 244)(12 346 245)(13 347 246)(14 348 247)(15 349 248)(16 350 249)(17 351 250)(18 352 251)(19 353 252)(20 354 253)(21 441 427)(22 442 428)(23 443 429)(24 444 430)(25 445 431)(26 446 432)(27 447 433)(28 448 434)(29 449 435)(30 450 436)(31 451 437)(32 452 438)(33 453 439)(34 454 440)(35 455 421)(36 456 422)(37 457 423)(38 458 424)(39 459 425)(40 460 426)(41 193 234)(42 194 235)(43 195 236)(44 196 237)(45 197 238)(46 198 239)(47 199 240)(48 200 221)(49 181 222)(50 182 223)(51 183 224)(52 184 225)(53 185 226)(54 186 227)(55 187 228)(56 188 229)(57 189 230)(58 190 231)(59 191 232)(60 192 233)(61 282 400)(62 283 381)(63 284 382)(64 285 383)(65 286 384)(66 287 385)(67 288 386)(68 289 387)(69 290 388)(70 291 389)(71 292 390)(72 293 391)(73 294 392)(74 295 393)(75 296 394)(76 297 395)(77 298 396)(78 299 397)(79 300 398)(80 281 399)(81 213 107)(82 214 108)(83 215 109)(84 216 110)(85 217 111)(86 218 112)(87 219 113)(88 220 114)(89 201 115)(90 202 116)(91 203 117)(92 204 118)(93 205 119)(94 206 120)(95 207 101)(96 208 102)(97 209 103)(98 210 104)(99 211 105)(100 212 106)(121 323 311)(122 324 312)(123 325 313)(124 326 314)(125 327 315)(126 328 316)(127 329 317)(128 330 318)(129 331 319)(130 332 320)(131 333 301)(132 334 302)(133 335 303)(134 336 304)(135 337 305)(136 338 306)(137 339 307)(138 340 308)(139 321 309)(140 322 310)(141 164 370)(142 165 371)(143 166 372)(144 167 373)(145 168 374)(146 169 375)(147 170 376)(148 171 377)(149 172 378)(150 173 379)(151 174 380)(152 175 361)(153 176 362)(154 177 363)(155 178 364)(156 179 365)(157 180 366)(158 161 367)(159 162 368)(160 163 369)(261 404 466)(262 405 467)(263 406 468)(264 407 469)(265 408 470)(266 409 471)(267 410 472)(268 411 473)(269 412 474)(270 413 475)(271 414 476)(272 415 477)(273 416 478)(274 417 479)(275 418 480)(276 419 461)(277 420 462)(278 401 463)(279 402 464)(280 403 465)
(1 421 232 82)(2 83 233 422)(3 423 234 84)(4 85 235 424)(5 425 236 86)(6 87 237 426)(7 427 238 88)(8 89 239 428)(9 429 240 90)(10 91 221 430)(11 431 222 92)(12 93 223 432)(13 433 224 94)(14 95 225 434)(15 435 226 96)(16 97 227 436)(17 437 228 98)(18 99 229 438)(19 439 230 100)(20 81 231 440)(21 45 220 341)(22 342 201 46)(23 47 202 343)(24 344 203 48)(25 49 204 345)(26 346 205 50)(27 51 206 347)(28 348 207 52)(29 53 208 349)(30 350 209 54)(31 55 210 351)(32 352 211 56)(33 57 212 353)(34 354 213 58)(35 59 214 355)(36 356 215 60)(37 41 216 357)(38 358 217 42)(39 43 218 359)(40 360 219 44)(61 420 122 361)(62 362 123 401)(63 402 124 363)(64 364 125 403)(65 404 126 365)(66 366 127 405)(67 406 128 367)(68 368 129 407)(69 408 130 369)(70 370 131 409)(71 410 132 371)(72 372 133 411)(73 412 134 373)(74 374 135 413)(75 414 136 375)(76 376 137 415)(77 416 138 377)(78 378 139 417)(79 418 140 379)(80 380 121 419)(101 184 448 247)(102 248 449 185)(103 186 450 249)(104 250 451 187)(105 188 452 251)(106 252 453 189)(107 190 454 253)(108 254 455 191)(109 192 456 255)(110 256 457 193)(111 194 458 257)(112 258 459 195)(113 196 460 259)(114 260 441 197)(115 198 442 241)(116 242 443 199)(117 200 444 243)(118 244 445 181)(119 182 446 245)(120 246 447 183)(141 333 471 291)(142 292 472 334)(143 335 473 293)(144 294 474 336)(145 337 475 295)(146 296 476 338)(147 339 477 297)(148 298 478 340)(149 321 479 299)(150 300 480 322)(151 323 461 281)(152 282 462 324)(153 325 463 283)(154 284 464 326)(155 327 465 285)(156 286 466 328)(157 329 467 287)(158 288 468 330)(159 331 469 289)(160 290 470 332)(161 386 263 318)(162 319 264 387)(163 388 265 320)(164 301 266 389)(165 390 267 302)(166 303 268 391)(167 392 269 304)(168 305 270 393)(169 394 271 306)(170 307 272 395)(171 396 273 308)(172 309 274 397)(173 398 275 310)(174 311 276 399)(175 400 277 312)(176 313 278 381)(177 382 279 314)(178 315 280 383)(179 384 261 316)(180 317 262 385)
(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)
(1 329 11 339)(2 286 12 296)(3 327 13 337)(4 284 14 294)(5 325 15 335)(6 282 16 292)(7 323 17 333)(8 300 18 290)(9 321 19 331)(10 298 20 288)(21 276 31 266)(22 173 32 163)(23 274 33 264)(24 171 34 161)(25 272 35 262)(26 169 36 179)(27 270 37 280)(28 167 38 177)(29 268 39 278)(30 165 40 175)(41 383 51 393)(42 314 52 304)(43 381 53 391)(44 312 54 302)(45 399 55 389)(46 310 56 320)(47 397 57 387)(48 308 58 318)(49 395 59 385)(50 306 60 316)(61 249 71 259)(62 185 72 195)(63 247 73 257)(64 183 74 193)(65 245 75 255)(66 181 76 191)(67 243 77 253)(68 199 78 189)(69 241 79 251)(70 197 80 187)(81 468 91 478)(82 157 92 147)(83 466 93 476)(84 155 94 145)(85 464 95 474)(86 153 96 143)(87 462 97 472)(88 151 98 141)(89 480 99 470)(90 149 100 159)(101 412 111 402)(102 372 112 362)(103 410 113 420)(104 370 114 380)(105 408 115 418)(106 368 116 378)(107 406 117 416)(108 366 118 376)(109 404 119 414)(110 364 120 374)(121 250 131 260)(122 186 132 196)(123 248 133 258)(124 184 134 194)(125 246 135 256)(126 182 136 192)(127 244 137 254)(128 200 138 190)(129 242 139 252)(130 198 140 188)(142 426 152 436)(144 424 154 434)(146 422 156 432)(148 440 158 430)(150 438 160 428)(162 202 172 212)(164 220 174 210)(166 218 176 208)(168 216 178 206)(170 214 180 204)(201 275 211 265)(203 273 213 263)(205 271 215 261)(207 269 217 279)(209 267 219 277)(221 340 231 330)(222 297 232 287)(223 338 233 328)(224 295 234 285)(225 336 235 326)(226 293 236 283)(227 334 237 324)(228 291 238 281)(229 332 239 322)(230 289 240 299)(301 341 311 351)(303 359 313 349)(305 357 315 347)(307 355 317 345)(309 353 319 343)(342 398 352 388)(344 396 354 386)(346 394 356 384)(348 392 358 382)(350 390 360 400)(361 450 371 460)(363 448 373 458)(365 446 375 456)(367 444 377 454)(369 442 379 452)(401 449 411 459)(403 447 413 457)(405 445 415 455)(407 443 417 453)(409 441 419 451)(421 467 431 477)(423 465 433 475)(425 463 435 473)(427 461 437 471)(429 479 439 469)

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

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

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

102 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H4I4J5A5B6A···6F10A···10F12A12B12C12D12E12F12G12H12I···12P12Q12R12S12T15A15B15C15D20A···20L30A···30L60A···60X
order1222334444444444556···610···10121212121212121212···12121212121515151520···2030···3060···60
size1111112244101010102020221···12···22222444410···102020202022224···42···24···4

102 irreducible representations

dim111111111122222222224444
type+++++-++--+
imageC1C2C2C2C2C3C6C6C6C6Q8D5C4○D4D10C3×Q8C3×D5Dic10C3×C4○D4C6×D5C3×Dic10D42D5Q82D5C3×D42D5C3×Q82D5
kernelC3×C4.Dic10C12×Dic5C3×C10.D4C3×C4⋊Dic5C15×C4⋊C4C4.Dic10C4×Dic5C10.D4C4⋊Dic5C5×C4⋊C4C60C3×C4⋊C4C30C2×C12C20C4⋊C4C12C10C2×C4C4C6C6C2C2
# reps11231224622246448812162244

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

47000
04700
0010
0001
,
60000
06000
00110
005450
,
253200
22700
00215
003440
,
55600
545600
0010
001660
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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