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G = C10×Dic3⋊C4order 480 = 25·3·5

Direct product of C10 and Dic3⋊C4

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

Aliases: C10×Dic3⋊C4, C3011(C4⋊C4), C6.7(Q8×C10), C6.38(D4×C10), C30.88(C2×Q8), (C2×C30).22Q8, (C2×Dic3)⋊4C20, Dic34(C2×C20), (C2×C30).178D4, (C2×C20).375D6, C30.421(C2×D4), (C22×C60).7C2, (C22×C20).8S3, C2.2(C10×Dic6), (C10×Dic3)⋊14C4, (C22×C12).4C10, C23.34(S3×C10), C22.16(S3×C20), C6.17(C22×C20), (C2×C10).16Dic6, C10.47(C2×Dic6), C22.4(C5×Dic6), C30.208(C22×C4), (C2×C30).420C23, (C2×C60).455C22, (C22×C10).149D6, (C22×Dic3).4C10, (C22×C30).171C22, (C10×Dic3).223C22, C61(C5×C4⋊C4), C32(C10×C4⋊C4), C1521(C2×C4⋊C4), C2.18(S3×C2×C20), (C2×C6).5(C5×Q8), C10.145(S3×C2×C4), (C2×C6).35(C5×D4), C2.1(C10×C3⋊D4), (C2×C6).17(C2×C20), (C2×C4).67(S3×C10), (C2×C10).88(C4×S3), (C22×C4).5(C5×S3), C22.20(S3×C2×C10), (C2×C12).74(C2×C10), (C2×C30).162(C2×C4), (C5×Dic3)⋊24(C2×C4), C10.123(C2×C3⋊D4), (Dic3×C2×C10).12C2, C22.19(C5×C3⋊D4), (C2×C10).91(C3⋊D4), (C22×C6).33(C2×C10), (C2×C6).41(C22×C10), (C2×C10).354(C22×S3), (C2×Dic3).31(C2×C10), SmallGroup(480,802)

Series: Derived Chief Lower central Upper central

C1C6 — C10×Dic3⋊C4
C1C3C6C2×C6C2×C30C10×Dic3Dic3×C2×C10 — C10×Dic3⋊C4
C3C6 — C10×Dic3⋊C4
C1C22×C10C22×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 [×3], C2 [×4], C3, C4 [×8], C22, C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×4], Dic3 [×4], Dic3 [×2], C12 [×2], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4, C22×C4 [×2], C20 [×8], C2×C10, C2×C10 [×6], C2×Dic3 [×8], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×4], C2×C4⋊C4, C2×C20 [×2], C2×C20 [×12], C22×C10, Dic3⋊C4 [×4], C22×Dic3 [×2], C22×C12, C5×Dic3 [×4], C5×Dic3 [×2], C60 [×2], C2×C30, C2×C30 [×6], C5×C4⋊C4 [×4], C22×C20, C22×C20 [×2], C2×Dic3⋊C4, C10×Dic3 [×8], C10×Dic3 [×2], C2×C60 [×2], C2×C60 [×2], C22×C30, C10×C4⋊C4, C5×Dic3⋊C4 [×4], Dic3×C2×C10 [×2], C22×C60, C10×Dic3⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, C10 [×7], D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C20 [×4], C2×C10 [×7], Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C5×S3, C2×C4⋊C4, C2×C20 [×6], C5×D4 [×2], C5×Q8 [×2], C22×C10, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, S3×C10 [×3], C5×C4⋊C4 [×4], C22×C20, D4×C10, Q8×C10, C2×Dic3⋊C4, C5×Dic6 [×2], S3×C20 [×2], C5×C3⋊D4 [×2], S3×C2×C10, C10×C4⋊C4, C5×Dic3⋊C4 [×4], C10×Dic6, S3×C2×C20, C10×C3⋊D4, C10×Dic3⋊C4

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

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

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

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

180 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L5A5B5C5D6A···6G10A···10AB12A···12H15A15B15C15D20A···20P20Q···20AV30A···30AB60A···60AF
order12···2344444···455556···610···1012···121515151520···2020···2030···3060···60
size11···1222226···611112···21···12···222222···26···62···22···2

180 irreducible representations

dim11111111112222222222222222
type++++++-++-
imageC1C2C2C2C4C5C10C10C10C20S3D4Q8D6D6Dic6C4×S3C3⋊D4C5×S3C5×D4C5×Q8S3×C10S3×C10C5×Dic6S3×C20C5×C3⋊D4
kernelC10×Dic3⋊C4C5×Dic3⋊C4Dic3×C2×C10C22×C60C10×Dic3C2×Dic3⋊C4Dic3⋊C4C22×Dic3C22×C12C2×Dic3C22×C20C2×C30C2×C30C2×C20C22×C10C2×C10C2×C10C2×C10C22×C4C2×C6C2×C6C2×C4C23C22C22C22
# reps1421841684321222144448884161616

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

60000000
0900000
0090000
0003000
0000300
0000010
0000001
,
1000000
06000000
00600000
00060000
00006000
00000060
00000160
,
1000000
013470000
047480000
000225300
000533900
000003451
000002427
,
11000000
0010000
06000000
0000100
00060000
00000600
00000060

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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