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## G = C10×Dic3⋊C4order 480 = 25·3·5

### Direct product of C10 and Dic3⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C10×Dic3⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — Dic3×C2×C10 — C10×Dic3⋊C4
 Lower central C3 — C6 — C10×Dic3⋊C4
 Upper central C1 — C22×C10 — C22×C20

Generators and relations for C10×Dic3⋊C4
G = < a,b,c,d | a10=b6=d4=1, c2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 324 in 184 conjugacy classes, 114 normal (34 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, C22×C4, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C30, C30, C2×C4⋊C4, C2×C20, C2×C20, C22×C10, Dic3⋊C4, C22×Dic3, C22×C12, C5×Dic3, C5×Dic3, C60, C2×C30, C2×C30, C5×C4⋊C4, C22×C20, C22×C20, C2×Dic3⋊C4, C10×Dic3, C10×Dic3, C2×C60, C2×C60, C22×C30, C10×C4⋊C4, C5×Dic3⋊C4, Dic3×C2×C10, C22×C60, C10×Dic3⋊C4
Quotients:

Smallest permutation representation of C10×Dic3⋊C4
Regular action on 480 points
Generators in S480
(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 72 294 103 219 136)(2 73 295 104 220 137)(3 74 296 105 211 138)(4 75 297 106 212 139)(5 76 298 107 213 140)(6 77 299 108 214 131)(7 78 300 109 215 132)(8 79 291 110 216 133)(9 80 292 101 217 134)(10 71 293 102 218 135)(11 45 461 336 451 170)(12 46 462 337 452 161)(13 47 463 338 453 162)(14 48 464 339 454 163)(15 49 465 340 455 164)(16 50 466 331 456 165)(17 41 467 332 457 166)(18 42 468 333 458 167)(19 43 469 334 459 168)(20 44 470 335 460 169)(21 146 306 324 449 156)(22 147 307 325 450 157)(23 148 308 326 441 158)(24 149 309 327 442 159)(25 150 310 328 443 160)(26 141 301 329 444 151)(27 142 302 330 445 152)(28 143 303 321 446 153)(29 144 304 322 447 154)(30 145 305 323 448 155)(31 122 360 258 373 480)(32 123 351 259 374 471)(33 124 352 260 375 472)(34 125 353 251 376 473)(35 126 354 252 377 474)(36 127 355 253 378 475)(37 128 356 254 379 476)(38 129 357 255 380 477)(39 130 358 256 371 478)(40 121 359 257 372 479)(51 188 114 283 370 234)(52 189 115 284 361 235)(53 190 116 285 362 236)(54 181 117 286 363 237)(55 182 118 287 364 238)(56 183 119 288 365 239)(57 184 120 289 366 240)(58 185 111 290 367 231)(59 186 112 281 368 232)(60 187 113 282 369 233)(61 226 385 270 242 92)(62 227 386 261 243 93)(63 228 387 262 244 94)(64 229 388 263 245 95)(65 230 389 264 246 96)(66 221 390 265 247 97)(67 222 381 266 248 98)(68 223 382 267 249 99)(69 224 383 268 250 100)(70 225 384 269 241 91)(81 191 397 272 407 206)(82 192 398 273 408 207)(83 193 399 274 409 208)(84 194 400 275 410 209)(85 195 391 276 401 210)(86 196 392 277 402 201)(87 197 393 278 403 202)(88 198 394 279 404 203)(89 199 395 280 405 204)(90 200 396 271 406 205)(171 312 437 414 430 343)(172 313 438 415 421 344)(173 314 439 416 422 345)(174 315 440 417 423 346)(175 316 431 418 424 347)(176 317 432 419 425 348)(177 318 433 420 426 349)(178 319 434 411 427 350)(179 320 435 412 428 341)(180 311 436 413 429 342)
(1 372 103 121)(2 373 104 122)(3 374 105 123)(4 375 106 124)(5 376 107 125)(6 377 108 126)(7 378 109 127)(8 379 110 128)(9 380 101 129)(10 371 102 130)(11 271 336 90)(12 272 337 81)(13 273 338 82)(14 274 339 83)(15 275 340 84)(16 276 331 85)(17 277 332 86)(18 278 333 87)(19 279 334 88)(20 280 335 89)(21 269 324 70)(22 270 325 61)(23 261 326 62)(24 262 327 63)(25 263 328 64)(26 264 329 65)(27 265 330 66)(28 266 321 67)(29 267 322 68)(30 268 323 69)(31 73 258 220)(32 74 259 211)(33 75 260 212)(34 76 251 213)(35 77 252 214)(36 78 253 215)(37 79 254 216)(38 80 255 217)(39 71 256 218)(40 72 257 219)(41 392 457 201)(42 393 458 202)(43 394 459 203)(44 395 460 204)(45 396 451 205)(46 397 452 206)(47 398 453 207)(48 399 454 208)(49 400 455 209)(50 391 456 210)(51 432 283 348)(52 433 284 349)(53 434 285 350)(54 435 286 341)(55 436 287 342)(56 437 288 343)(57 438 289 344)(58 439 290 345)(59 440 281 346)(60 431 282 347)(91 146 384 449)(92 147 385 450)(93 148 386 441)(94 149 387 442)(95 150 388 443)(96 141 389 444)(97 142 390 445)(98 143 381 446)(99 144 382 447)(100 145 383 448)(111 173 231 416)(112 174 232 417)(113 175 233 418)(114 176 234 419)(115 177 235 420)(116 178 236 411)(117 179 237 412)(118 180 238 413)(119 171 239 414)(120 172 240 415)(131 474 299 354)(132 475 300 355)(133 476 291 356)(134 477 292 357)(135 478 293 358)(136 479 294 359)(137 480 295 360)(138 471 296 351)(139 472 297 352)(140 473 298 353)(151 246 301 230)(152 247 302 221)(153 248 303 222)(154 249 304 223)(155 250 305 224)(156 241 306 225)(157 242 307 226)(158 243 308 227)(159 244 309 228)(160 245 310 229)(161 407 462 191)(162 408 463 192)(163 409 464 193)(164 410 465 194)(165 401 466 195)(166 402 467 196)(167 403 468 197)(168 404 469 198)(169 405 470 199)(170 406 461 200)(181 320 363 428)(182 311 364 429)(183 312 365 430)(184 313 366 421)(185 314 367 422)(186 315 368 423)(187 316 369 424)(188 317 370 425)(189 318 361 426)(190 319 362 427)
(1 385 288 391)(2 386 289 392)(3 387 290 393)(4 388 281 394)(5 389 282 395)(6 390 283 396)(7 381 284 397)(8 382 285 398)(9 383 286 399)(10 384 287 400)(11 252 27 425)(12 253 28 426)(13 254 29 427)(14 255 30 428)(15 256 21 429)(16 257 22 430)(17 258 23 421)(18 259 24 422)(19 260 25 423)(20 251 26 424)(31 326 313 332)(32 327 314 333)(33 328 315 334)(34 329 316 335)(35 330 317 336)(36 321 318 337)(37 322 319 338)(38 323 320 339)(39 324 311 340)(40 325 312 331)(41 373 148 344)(42 374 149 345)(43 375 150 346)(44 376 141 347)(45 377 142 348)(46 378 143 349)(47 379 144 350)(48 380 145 341)(49 371 146 342)(50 372 147 343)(51 205 108 97)(52 206 109 98)(53 207 110 99)(54 208 101 100)(55 209 102 91)(56 210 103 92)(57 201 104 93)(58 202 105 94)(59 203 106 95)(60 204 107 96)(61 183 85 219)(62 184 86 220)(63 185 87 211)(64 186 88 212)(65 187 89 213)(66 188 90 214)(67 189 81 215)(68 190 82 216)(69 181 83 217)(70 182 84 218)(71 269 364 275)(72 270 365 276)(73 261 366 277)(74 262 367 278)(75 263 368 279)(76 264 369 280)(77 265 370 271)(78 266 361 272)(79 267 362 273)(80 268 363 274)(111 197 138 228)(112 198 139 229)(113 199 140 230)(114 200 131 221)(115 191 132 222)(116 192 133 223)(117 193 134 224)(118 194 135 225)(119 195 136 226)(120 196 137 227)(121 450 437 456)(122 441 438 457)(123 442 439 458)(124 443 440 459)(125 444 431 460)(126 445 432 451)(127 446 433 452)(128 447 434 453)(129 448 435 454)(130 449 436 455)(151 418 169 353)(152 419 170 354)(153 420 161 355)(154 411 162 356)(155 412 163 357)(156 413 164 358)(157 414 165 359)(158 415 166 360)(159 416 167 351)(160 417 168 352)(171 466 479 307)(172 467 480 308)(173 468 471 309)(174 469 472 310)(175 470 473 301)(176 461 474 302)(177 462 475 303)(178 463 476 304)(179 464 477 305)(180 465 478 306)(231 403 296 244)(232 404 297 245)(233 405 298 246)(234 406 299 247)(235 407 300 248)(236 408 291 249)(237 409 292 250)(238 410 293 241)(239 401 294 242)(240 402 295 243)

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

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

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

180 conjugacy classes

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

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + + - image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 S3 D4 Q8 D6 D6 Dic6 C4×S3 C3⋊D4 C5×S3 C5×D4 C5×Q8 S3×C10 S3×C10 C5×Dic6 S3×C20 C5×C3⋊D4 kernel C10×Dic3⋊C4 C5×Dic3⋊C4 Dic3×C2×C10 C22×C60 C10×Dic3 C2×Dic3⋊C4 Dic3⋊C4 C22×Dic3 C22×C12 C2×Dic3 C22×C20 C2×C30 C2×C30 C2×C20 C22×C10 C2×C10 C2×C10 C2×C10 C22×C4 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 4 2 1 8 4 16 8 4 32 1 2 2 2 1 4 4 4 4 8 8 8 4 16 16 16

Matrix representation of C10×Dic3⋊C4 in GL7(𝔽61)

 60 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 1 60
,
 1 0 0 0 0 0 0 0 13 47 0 0 0 0 0 47 48 0 0 0 0 0 0 0 22 53 0 0 0 0 0 53 39 0 0 0 0 0 0 0 34 51 0 0 0 0 0 24 27
,
 11 0 0 0 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60

G:=sub<GL(7,GF(61))| [60,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,60,60],[1,0,0,0,0,0,0,0,13,47,0,0,0,0,0,47,48,0,0,0,0,0,0,0,22,53,0,0,0,0,0,53,39,0,0,0,0,0,0,0,34,24,0,0,0,0,0,51,27],[11,0,0,0,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60] >;

C10×Dic3⋊C4 in GAP, Magma, Sage, TeX

C_{10}\times {\rm Dic}_3\rtimes C_4
% in TeX

G:=Group("C10xDic3:C4");
// GroupNames label

G:=SmallGroup(480,802);
// by ID

G=gap.SmallGroup(480,802);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,1766,226,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^6=d^4=1,c^2=b^3,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^3*c>;
// generators/relations

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