direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10×C4⋊Dic3, C60⋊42(C2×C4), (C2×C60)⋊24C4, C12⋊7(C2×C20), (C2×C12)⋊5C20, C30⋊12(C4⋊C4), C6.9(Q8×C10), C4⋊2(C10×Dic3), C6.15(D4×C10), C2.2(C10×D12), (C2×C30).23Q8, C30.90(C2×Q8), C20⋊12(C2×Dic3), (C2×C20)⋊13Dic3, (C2×C10).56D12, C30.302(C2×D4), C10.86(C2×D12), (C2×C30).127D4, (C2×C20).438D6, C2.3(C10×Dic6), (C22×C60).23C2, C6.23(C22×C20), (C22×C12).7C10, (C22×C20).20S3, C23.35(S3×C10), (C2×C10).17Dic6, C10.49(C2×Dic6), C22.15(C5×D12), C22.5(C5×Dic6), (C2×C30).422C23, (C2×C60).531C22, C30.230(C22×C4), (C22×C10).150D6, (C22×Dic3).5C10, C10.46(C22×Dic3), C22.14(C10×Dic3), (C22×C30).173C22, (C10×Dic3).224C22, C6⋊2(C5×C4⋊C4), C3⋊3(C10×C4⋊C4), C15⋊22(C2×C4⋊C4), (C2×C6).6(C5×Q8), (C2×C4)⋊3(C5×Dic3), (C2×C6).20(C5×D4), C2.4(Dic3×C2×C10), (C2×C6).34(C2×C20), (C2×C4).86(S3×C10), C22.21(S3×C2×C10), (C22×C4).8(C5×S3), (C2×C30).202(C2×C4), (C2×C12).97(C2×C10), (Dic3×C2×C10).13C2, (C22×C6).35(C2×C10), (C2×C6).43(C22×C10), (C2×C10).65(C2×Dic3), (C2×C10).356(C22×S3), (C2×Dic3).32(C2×C10), SmallGroup(480,804)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C4⋊Dic3
G = < a,b,c,d | a10=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 324 in 184 conjugacy classes, 130 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C30, C30, C2×C4⋊C4, C2×C20, C2×C20, C22×C10, C4⋊Dic3, C22×Dic3, C22×C12, C5×Dic3, C60, C2×C30, C2×C30, C5×C4⋊C4, C22×C20, C22×C20, C2×C4⋊Dic3, C10×Dic3, C10×Dic3, C2×C60, C22×C30, C10×C4⋊C4, C5×C4⋊Dic3, Dic3×C2×C10, C22×C60, C10×C4⋊Dic3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, Q8, C23, C10, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C20, C2×C10, Dic6, D12, C2×Dic3, C22×S3, C5×S3, C2×C4⋊C4, C2×C20, C5×D4, C5×Q8, C22×C10, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, C5×Dic3, S3×C10, C5×C4⋊C4, C22×C20, D4×C10, Q8×C10, C2×C4⋊Dic3, C5×Dic6, C5×D12, C10×Dic3, S3×C2×C10, C10×C4⋊C4, C5×C4⋊Dic3, C10×Dic6, C10×D12, Dic3×C2×C10, C10×C4⋊Dic3
►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)
(1 212 188 296)(2 213 189 297)(3 214 190 298)(4 215 181 299)(5 216 182 300)(6 217 183 291)(7 218 184 292)(8 219 185 293)(9 220 186 294)(10 211 187 295)(11 23 250 358)(12 24 241 359)(13 25 242 360)(14 26 243 351)(15 27 244 352)(16 28 245 353)(17 29 246 354)(18 30 247 355)(19 21 248 356)(20 22 249 357)(31 342 463 319)(32 343 464 320)(33 344 465 311)(34 345 466 312)(35 346 467 313)(36 347 468 314)(37 348 469 315)(38 349 470 316)(39 350 461 317)(40 341 462 318)(41 433 445 472)(42 434 446 473)(43 435 447 474)(44 436 448 475)(45 437 449 476)(46 438 450 477)(47 439 441 478)(48 440 442 479)(49 431 443 480)(50 432 444 471)(51 399 365 87)(52 400 366 88)(53 391 367 89)(54 392 368 90)(55 393 369 81)(56 394 370 82)(57 395 361 83)(58 396 362 84)(59 397 363 85)(60 398 364 86)(61 137 404 283)(62 138 405 284)(63 139 406 285)(64 140 407 286)(65 131 408 287)(66 132 409 288)(67 133 410 289)(68 134 401 290)(69 135 402 281)(70 136 403 282)(71 225 104 263)(72 226 105 264)(73 227 106 265)(74 228 107 266)(75 229 108 267)(76 230 109 268)(77 221 110 269)(78 222 101 270)(79 223 102 261)(80 224 103 262)(91 116 385 373)(92 117 386 374)(93 118 387 375)(94 119 388 376)(95 120 389 377)(96 111 390 378)(97 112 381 379)(98 113 382 380)(99 114 383 371)(100 115 384 372)(121 143 255 327)(122 144 256 328)(123 145 257 329)(124 146 258 330)(125 147 259 321)(126 148 260 322)(127 149 251 323)(128 150 252 324)(129 141 253 325)(130 142 254 326)(151 418 337 163)(152 419 338 164)(153 420 339 165)(154 411 340 166)(155 412 331 167)(156 413 332 168)(157 414 333 169)(158 415 334 170)(159 416 335 161)(160 417 336 162)(171 426 460 306)(172 427 451 307)(173 428 452 308)(174 429 453 309)(175 430 454 310)(176 421 455 301)(177 422 456 302)(178 423 457 303)(179 424 458 304)(180 425 459 305)(191 279 240 203)(192 280 231 204)(193 271 232 205)(194 272 233 206)(195 273 234 207)(196 274 235 208)(197 275 236 209)(198 276 237 210)(199 277 238 201)(200 278 239 202)
(1 134 375 363 105 239)(2 135 376 364 106 240)(3 136 377 365 107 231)(4 137 378 366 108 232)(5 138 379 367 109 233)(6 139 380 368 110 234)(7 140 371 369 101 235)(8 131 372 370 102 236)(9 132 373 361 103 237)(10 133 374 362 104 238)(11 166 254 305 476 342)(12 167 255 306 477 343)(13 168 256 307 478 344)(14 169 257 308 479 345)(15 170 258 309 480 346)(16 161 259 310 471 347)(17 162 260 301 472 348)(18 163 251 302 473 349)(19 164 252 303 474 350)(20 165 253 304 475 341)(21 152 324 178 43 461)(22 153 325 179 44 462)(23 154 326 180 45 463)(24 155 327 171 46 464)(25 156 328 172 47 465)(26 157 329 173 48 466)(27 158 330 174 49 467)(28 159 321 175 50 468)(29 160 322 176 41 469)(30 151 323 177 42 470)(31 358 340 142 459 449)(32 359 331 143 460 450)(33 360 332 144 451 441)(34 351 333 145 452 442)(35 352 334 146 453 443)(36 353 335 147 454 444)(37 354 336 148 455 445)(38 355 337 149 456 446)(39 356 338 150 457 447)(40 357 339 141 458 448)(51 74 192 190 282 120)(52 75 193 181 283 111)(53 76 194 182 284 112)(54 77 195 183 285 113)(55 78 196 184 286 114)(56 79 197 185 287 115)(57 80 198 186 288 116)(58 71 199 187 289 117)(59 72 200 188 290 118)(60 73 191 189 281 119)(61 390 400 229 271 299)(62 381 391 230 272 300)(63 382 392 221 273 291)(64 383 393 222 274 292)(65 384 394 223 275 293)(66 385 395 224 276 294)(67 386 396 225 277 295)(68 387 397 226 278 296)(69 388 398 227 279 297)(70 389 399 228 280 298)(81 270 208 218 407 99)(82 261 209 219 408 100)(83 262 210 220 409 91)(84 263 201 211 410 92)(85 264 202 212 401 93)(86 265 203 213 402 94)(87 266 204 214 403 95)(88 267 205 215 404 96)(89 268 206 216 405 97)(90 269 207 217 406 98)(121 426 438 320 241 412)(122 427 439 311 242 413)(123 428 440 312 243 414)(124 429 431 313 244 415)(125 430 432 314 245 416)(126 421 433 315 246 417)(127 422 434 316 247 418)(128 423 435 317 248 419)(129 424 436 318 249 420)(130 425 437 319 250 411)
(1 14 363 308)(2 15 364 309)(3 16 365 310)(4 17 366 301)(5 18 367 302)(6 19 368 303)(7 20 369 304)(8 11 370 305)(9 12 361 306)(10 13 362 307)(21 392 178 291)(22 393 179 292)(23 394 180 293)(24 395 171 294)(25 396 172 295)(26 397 173 296)(27 398 174 297)(28 399 175 298)(29 400 176 299)(30 391 177 300)(31 261 142 408)(32 262 143 409)(33 263 144 410)(34 264 145 401)(35 265 146 402)(36 266 147 403)(37 267 148 404)(38 268 149 405)(39 269 150 406)(40 270 141 407)(41 271 160 390)(42 272 151 381)(43 273 152 382)(44 274 153 383)(45 275 154 384)(46 276 155 385)(47 277 156 386)(48 278 157 387)(49 279 158 388)(50 280 159 389)(51 430 190 245)(52 421 181 246)(53 422 182 247)(54 423 183 248)(55 424 184 249)(56 425 185 250)(57 426 186 241)(58 427 187 242)(59 428 188 243)(60 429 189 244)(61 469 229 322)(62 470 230 323)(63 461 221 324)(64 462 222 325)(65 463 223 326)(66 464 224 327)(67 465 225 328)(68 466 226 329)(69 467 227 330)(70 468 228 321)(71 122 289 311)(72 123 290 312)(73 124 281 313)(74 125 282 314)(75 126 283 315)(76 127 284 316)(77 128 285 317)(78 129 286 318)(79 130 287 319)(80 121 288 320)(81 458 218 357)(82 459 219 358)(83 460 220 359)(84 451 211 360)(85 452 212 351)(86 453 213 352)(87 454 214 353)(88 455 215 354)(89 456 216 355)(90 457 217 356)(91 450 210 331)(92 441 201 332)(93 442 202 333)(94 443 203 334)(95 444 204 335)(96 445 205 336)(97 446 206 337)(98 447 207 338)(99 448 208 339)(100 449 209 340)(101 253 140 341)(102 254 131 342)(103 255 132 343)(104 256 133 344)(105 257 134 345)(106 258 135 346)(107 259 136 347)(108 260 137 348)(109 251 138 349)(110 252 139 350)(111 433 193 417)(112 434 194 418)(113 435 195 419)(114 436 196 420)(115 437 197 411)(116 438 198 412)(117 439 199 413)(118 440 200 414)(119 431 191 415)(120 432 192 416)(161 377 471 231)(162 378 472 232)(163 379 473 233)(164 380 474 234)(165 371 475 235)(166 372 476 236)(167 373 477 237)(168 374 478 238)(169 375 479 239)(170 376 480 240)
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), (1,212,188,296)(2,213,189,297)(3,214,190,298)(4,215,181,299)(5,216,182,300)(6,217,183,291)(7,218,184,292)(8,219,185,293)(9,220,186,294)(10,211,187,295)(11,23,250,358)(12,24,241,359)(13,25,242,360)(14,26,243,351)(15,27,244,352)(16,28,245,353)(17,29,246,354)(18,30,247,355)(19,21,248,356)(20,22,249,357)(31,342,463,319)(32,343,464,320)(33,344,465,311)(34,345,466,312)(35,346,467,313)(36,347,468,314)(37,348,469,315)(38,349,470,316)(39,350,461,317)(40,341,462,318)(41,433,445,472)(42,434,446,473)(43,435,447,474)(44,436,448,475)(45,437,449,476)(46,438,450,477)(47,439,441,478)(48,440,442,479)(49,431,443,480)(50,432,444,471)(51,399,365,87)(52,400,366,88)(53,391,367,89)(54,392,368,90)(55,393,369,81)(56,394,370,82)(57,395,361,83)(58,396,362,84)(59,397,363,85)(60,398,364,86)(61,137,404,283)(62,138,405,284)(63,139,406,285)(64,140,407,286)(65,131,408,287)(66,132,409,288)(67,133,410,289)(68,134,401,290)(69,135,402,281)(70,136,403,282)(71,225,104,263)(72,226,105,264)(73,227,106,265)(74,228,107,266)(75,229,108,267)(76,230,109,268)(77,221,110,269)(78,222,101,270)(79,223,102,261)(80,224,103,262)(91,116,385,373)(92,117,386,374)(93,118,387,375)(94,119,388,376)(95,120,389,377)(96,111,390,378)(97,112,381,379)(98,113,382,380)(99,114,383,371)(100,115,384,372)(121,143,255,327)(122,144,256,328)(123,145,257,329)(124,146,258,330)(125,147,259,321)(126,148,260,322)(127,149,251,323)(128,150,252,324)(129,141,253,325)(130,142,254,326)(151,418,337,163)(152,419,338,164)(153,420,339,165)(154,411,340,166)(155,412,331,167)(156,413,332,168)(157,414,333,169)(158,415,334,170)(159,416,335,161)(160,417,336,162)(171,426,460,306)(172,427,451,307)(173,428,452,308)(174,429,453,309)(175,430,454,310)(176,421,455,301)(177,422,456,302)(178,423,457,303)(179,424,458,304)(180,425,459,305)(191,279,240,203)(192,280,231,204)(193,271,232,205)(194,272,233,206)(195,273,234,207)(196,274,235,208)(197,275,236,209)(198,276,237,210)(199,277,238,201)(200,278,239,202), (1,134,375,363,105,239)(2,135,376,364,106,240)(3,136,377,365,107,231)(4,137,378,366,108,232)(5,138,379,367,109,233)(6,139,380,368,110,234)(7,140,371,369,101,235)(8,131,372,370,102,236)(9,132,373,361,103,237)(10,133,374,362,104,238)(11,166,254,305,476,342)(12,167,255,306,477,343)(13,168,256,307,478,344)(14,169,257,308,479,345)(15,170,258,309,480,346)(16,161,259,310,471,347)(17,162,260,301,472,348)(18,163,251,302,473,349)(19,164,252,303,474,350)(20,165,253,304,475,341)(21,152,324,178,43,461)(22,153,325,179,44,462)(23,154,326,180,45,463)(24,155,327,171,46,464)(25,156,328,172,47,465)(26,157,329,173,48,466)(27,158,330,174,49,467)(28,159,321,175,50,468)(29,160,322,176,41,469)(30,151,323,177,42,470)(31,358,340,142,459,449)(32,359,331,143,460,450)(33,360,332,144,451,441)(34,351,333,145,452,442)(35,352,334,146,453,443)(36,353,335,147,454,444)(37,354,336,148,455,445)(38,355,337,149,456,446)(39,356,338,150,457,447)(40,357,339,141,458,448)(51,74,192,190,282,120)(52,75,193,181,283,111)(53,76,194,182,284,112)(54,77,195,183,285,113)(55,78,196,184,286,114)(56,79,197,185,287,115)(57,80,198,186,288,116)(58,71,199,187,289,117)(59,72,200,188,290,118)(60,73,191,189,281,119)(61,390,400,229,271,299)(62,381,391,230,272,300)(63,382,392,221,273,291)(64,383,393,222,274,292)(65,384,394,223,275,293)(66,385,395,224,276,294)(67,386,396,225,277,295)(68,387,397,226,278,296)(69,388,398,227,279,297)(70,389,399,228,280,298)(81,270,208,218,407,99)(82,261,209,219,408,100)(83,262,210,220,409,91)(84,263,201,211,410,92)(85,264,202,212,401,93)(86,265,203,213,402,94)(87,266,204,214,403,95)(88,267,205,215,404,96)(89,268,206,216,405,97)(90,269,207,217,406,98)(121,426,438,320,241,412)(122,427,439,311,242,413)(123,428,440,312,243,414)(124,429,431,313,244,415)(125,430,432,314,245,416)(126,421,433,315,246,417)(127,422,434,316,247,418)(128,423,435,317,248,419)(129,424,436,318,249,420)(130,425,437,319,250,411), (1,14,363,308)(2,15,364,309)(3,16,365,310)(4,17,366,301)(5,18,367,302)(6,19,368,303)(7,20,369,304)(8,11,370,305)(9,12,361,306)(10,13,362,307)(21,392,178,291)(22,393,179,292)(23,394,180,293)(24,395,171,294)(25,396,172,295)(26,397,173,296)(27,398,174,297)(28,399,175,298)(29,400,176,299)(30,391,177,300)(31,261,142,408)(32,262,143,409)(33,263,144,410)(34,264,145,401)(35,265,146,402)(36,266,147,403)(37,267,148,404)(38,268,149,405)(39,269,150,406)(40,270,141,407)(41,271,160,390)(42,272,151,381)(43,273,152,382)(44,274,153,383)(45,275,154,384)(46,276,155,385)(47,277,156,386)(48,278,157,387)(49,279,158,388)(50,280,159,389)(51,430,190,245)(52,421,181,246)(53,422,182,247)(54,423,183,248)(55,424,184,249)(56,425,185,250)(57,426,186,241)(58,427,187,242)(59,428,188,243)(60,429,189,244)(61,469,229,322)(62,470,230,323)(63,461,221,324)(64,462,222,325)(65,463,223,326)(66,464,224,327)(67,465,225,328)(68,466,226,329)(69,467,227,330)(70,468,228,321)(71,122,289,311)(72,123,290,312)(73,124,281,313)(74,125,282,314)(75,126,283,315)(76,127,284,316)(77,128,285,317)(78,129,286,318)(79,130,287,319)(80,121,288,320)(81,458,218,357)(82,459,219,358)(83,460,220,359)(84,451,211,360)(85,452,212,351)(86,453,213,352)(87,454,214,353)(88,455,215,354)(89,456,216,355)(90,457,217,356)(91,450,210,331)(92,441,201,332)(93,442,202,333)(94,443,203,334)(95,444,204,335)(96,445,205,336)(97,446,206,337)(98,447,207,338)(99,448,208,339)(100,449,209,340)(101,253,140,341)(102,254,131,342)(103,255,132,343)(104,256,133,344)(105,257,134,345)(106,258,135,346)(107,259,136,347)(108,260,137,348)(109,251,138,349)(110,252,139,350)(111,433,193,417)(112,434,194,418)(113,435,195,419)(114,436,196,420)(115,437,197,411)(116,438,198,412)(117,439,199,413)(118,440,200,414)(119,431,191,415)(120,432,192,416)(161,377,471,231)(162,378,472,232)(163,379,473,233)(164,380,474,234)(165,371,475,235)(166,372,476,236)(167,373,477,237)(168,374,478,238)(169,375,479,239)(170,376,480,240)>;
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), (1,212,188,296)(2,213,189,297)(3,214,190,298)(4,215,181,299)(5,216,182,300)(6,217,183,291)(7,218,184,292)(8,219,185,293)(9,220,186,294)(10,211,187,295)(11,23,250,358)(12,24,241,359)(13,25,242,360)(14,26,243,351)(15,27,244,352)(16,28,245,353)(17,29,246,354)(18,30,247,355)(19,21,248,356)(20,22,249,357)(31,342,463,319)(32,343,464,320)(33,344,465,311)(34,345,466,312)(35,346,467,313)(36,347,468,314)(37,348,469,315)(38,349,470,316)(39,350,461,317)(40,341,462,318)(41,433,445,472)(42,434,446,473)(43,435,447,474)(44,436,448,475)(45,437,449,476)(46,438,450,477)(47,439,441,478)(48,440,442,479)(49,431,443,480)(50,432,444,471)(51,399,365,87)(52,400,366,88)(53,391,367,89)(54,392,368,90)(55,393,369,81)(56,394,370,82)(57,395,361,83)(58,396,362,84)(59,397,363,85)(60,398,364,86)(61,137,404,283)(62,138,405,284)(63,139,406,285)(64,140,407,286)(65,131,408,287)(66,132,409,288)(67,133,410,289)(68,134,401,290)(69,135,402,281)(70,136,403,282)(71,225,104,263)(72,226,105,264)(73,227,106,265)(74,228,107,266)(75,229,108,267)(76,230,109,268)(77,221,110,269)(78,222,101,270)(79,223,102,261)(80,224,103,262)(91,116,385,373)(92,117,386,374)(93,118,387,375)(94,119,388,376)(95,120,389,377)(96,111,390,378)(97,112,381,379)(98,113,382,380)(99,114,383,371)(100,115,384,372)(121,143,255,327)(122,144,256,328)(123,145,257,329)(124,146,258,330)(125,147,259,321)(126,148,260,322)(127,149,251,323)(128,150,252,324)(129,141,253,325)(130,142,254,326)(151,418,337,163)(152,419,338,164)(153,420,339,165)(154,411,340,166)(155,412,331,167)(156,413,332,168)(157,414,333,169)(158,415,334,170)(159,416,335,161)(160,417,336,162)(171,426,460,306)(172,427,451,307)(173,428,452,308)(174,429,453,309)(175,430,454,310)(176,421,455,301)(177,422,456,302)(178,423,457,303)(179,424,458,304)(180,425,459,305)(191,279,240,203)(192,280,231,204)(193,271,232,205)(194,272,233,206)(195,273,234,207)(196,274,235,208)(197,275,236,209)(198,276,237,210)(199,277,238,201)(200,278,239,202), (1,134,375,363,105,239)(2,135,376,364,106,240)(3,136,377,365,107,231)(4,137,378,366,108,232)(5,138,379,367,109,233)(6,139,380,368,110,234)(7,140,371,369,101,235)(8,131,372,370,102,236)(9,132,373,361,103,237)(10,133,374,362,104,238)(11,166,254,305,476,342)(12,167,255,306,477,343)(13,168,256,307,478,344)(14,169,257,308,479,345)(15,170,258,309,480,346)(16,161,259,310,471,347)(17,162,260,301,472,348)(18,163,251,302,473,349)(19,164,252,303,474,350)(20,165,253,304,475,341)(21,152,324,178,43,461)(22,153,325,179,44,462)(23,154,326,180,45,463)(24,155,327,171,46,464)(25,156,328,172,47,465)(26,157,329,173,48,466)(27,158,330,174,49,467)(28,159,321,175,50,468)(29,160,322,176,41,469)(30,151,323,177,42,470)(31,358,340,142,459,449)(32,359,331,143,460,450)(33,360,332,144,451,441)(34,351,333,145,452,442)(35,352,334,146,453,443)(36,353,335,147,454,444)(37,354,336,148,455,445)(38,355,337,149,456,446)(39,356,338,150,457,447)(40,357,339,141,458,448)(51,74,192,190,282,120)(52,75,193,181,283,111)(53,76,194,182,284,112)(54,77,195,183,285,113)(55,78,196,184,286,114)(56,79,197,185,287,115)(57,80,198,186,288,116)(58,71,199,187,289,117)(59,72,200,188,290,118)(60,73,191,189,281,119)(61,390,400,229,271,299)(62,381,391,230,272,300)(63,382,392,221,273,291)(64,383,393,222,274,292)(65,384,394,223,275,293)(66,385,395,224,276,294)(67,386,396,225,277,295)(68,387,397,226,278,296)(69,388,398,227,279,297)(70,389,399,228,280,298)(81,270,208,218,407,99)(82,261,209,219,408,100)(83,262,210,220,409,91)(84,263,201,211,410,92)(85,264,202,212,401,93)(86,265,203,213,402,94)(87,266,204,214,403,95)(88,267,205,215,404,96)(89,268,206,216,405,97)(90,269,207,217,406,98)(121,426,438,320,241,412)(122,427,439,311,242,413)(123,428,440,312,243,414)(124,429,431,313,244,415)(125,430,432,314,245,416)(126,421,433,315,246,417)(127,422,434,316,247,418)(128,423,435,317,248,419)(129,424,436,318,249,420)(130,425,437,319,250,411), (1,14,363,308)(2,15,364,309)(3,16,365,310)(4,17,366,301)(5,18,367,302)(6,19,368,303)(7,20,369,304)(8,11,370,305)(9,12,361,306)(10,13,362,307)(21,392,178,291)(22,393,179,292)(23,394,180,293)(24,395,171,294)(25,396,172,295)(26,397,173,296)(27,398,174,297)(28,399,175,298)(29,400,176,299)(30,391,177,300)(31,261,142,408)(32,262,143,409)(33,263,144,410)(34,264,145,401)(35,265,146,402)(36,266,147,403)(37,267,148,404)(38,268,149,405)(39,269,150,406)(40,270,141,407)(41,271,160,390)(42,272,151,381)(43,273,152,382)(44,274,153,383)(45,275,154,384)(46,276,155,385)(47,277,156,386)(48,278,157,387)(49,279,158,388)(50,280,159,389)(51,430,190,245)(52,421,181,246)(53,422,182,247)(54,423,183,248)(55,424,184,249)(56,425,185,250)(57,426,186,241)(58,427,187,242)(59,428,188,243)(60,429,189,244)(61,469,229,322)(62,470,230,323)(63,461,221,324)(64,462,222,325)(65,463,223,326)(66,464,224,327)(67,465,225,328)(68,466,226,329)(69,467,227,330)(70,468,228,321)(71,122,289,311)(72,123,290,312)(73,124,281,313)(74,125,282,314)(75,126,283,315)(76,127,284,316)(77,128,285,317)(78,129,286,318)(79,130,287,319)(80,121,288,320)(81,458,218,357)(82,459,219,358)(83,460,220,359)(84,451,211,360)(85,452,212,351)(86,453,213,352)(87,454,214,353)(88,455,215,354)(89,456,216,355)(90,457,217,356)(91,450,210,331)(92,441,201,332)(93,442,202,333)(94,443,203,334)(95,444,204,335)(96,445,205,336)(97,446,206,337)(98,447,207,338)(99,448,208,339)(100,449,209,340)(101,253,140,341)(102,254,131,342)(103,255,132,343)(104,256,133,344)(105,257,134,345)(106,258,135,346)(107,259,136,347)(108,260,137,348)(109,251,138,349)(110,252,139,350)(111,433,193,417)(112,434,194,418)(113,435,195,419)(114,436,196,420)(115,437,197,411)(116,438,198,412)(117,439,199,413)(118,440,200,414)(119,431,191,415)(120,432,192,416)(161,377,471,231)(162,378,472,232)(163,379,473,233)(164,380,474,234)(165,371,475,235)(166,372,476,236)(167,373,477,237)(168,374,478,238)(169,375,479,239)(170,376,480,240) );
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)], [(1,212,188,296),(2,213,189,297),(3,214,190,298),(4,215,181,299),(5,216,182,300),(6,217,183,291),(7,218,184,292),(8,219,185,293),(9,220,186,294),(10,211,187,295),(11,23,250,358),(12,24,241,359),(13,25,242,360),(14,26,243,351),(15,27,244,352),(16,28,245,353),(17,29,246,354),(18,30,247,355),(19,21,248,356),(20,22,249,357),(31,342,463,319),(32,343,464,320),(33,344,465,311),(34,345,466,312),(35,346,467,313),(36,347,468,314),(37,348,469,315),(38,349,470,316),(39,350,461,317),(40,341,462,318),(41,433,445,472),(42,434,446,473),(43,435,447,474),(44,436,448,475),(45,437,449,476),(46,438,450,477),(47,439,441,478),(48,440,442,479),(49,431,443,480),(50,432,444,471),(51,399,365,87),(52,400,366,88),(53,391,367,89),(54,392,368,90),(55,393,369,81),(56,394,370,82),(57,395,361,83),(58,396,362,84),(59,397,363,85),(60,398,364,86),(61,137,404,283),(62,138,405,284),(63,139,406,285),(64,140,407,286),(65,131,408,287),(66,132,409,288),(67,133,410,289),(68,134,401,290),(69,135,402,281),(70,136,403,282),(71,225,104,263),(72,226,105,264),(73,227,106,265),(74,228,107,266),(75,229,108,267),(76,230,109,268),(77,221,110,269),(78,222,101,270),(79,223,102,261),(80,224,103,262),(91,116,385,373),(92,117,386,374),(93,118,387,375),(94,119,388,376),(95,120,389,377),(96,111,390,378),(97,112,381,379),(98,113,382,380),(99,114,383,371),(100,115,384,372),(121,143,255,327),(122,144,256,328),(123,145,257,329),(124,146,258,330),(125,147,259,321),(126,148,260,322),(127,149,251,323),(128,150,252,324),(129,141,253,325),(130,142,254,326),(151,418,337,163),(152,419,338,164),(153,420,339,165),(154,411,340,166),(155,412,331,167),(156,413,332,168),(157,414,333,169),(158,415,334,170),(159,416,335,161),(160,417,336,162),(171,426,460,306),(172,427,451,307),(173,428,452,308),(174,429,453,309),(175,430,454,310),(176,421,455,301),(177,422,456,302),(178,423,457,303),(179,424,458,304),(180,425,459,305),(191,279,240,203),(192,280,231,204),(193,271,232,205),(194,272,233,206),(195,273,234,207),(196,274,235,208),(197,275,236,209),(198,276,237,210),(199,277,238,201),(200,278,239,202)], [(1,134,375,363,105,239),(2,135,376,364,106,240),(3,136,377,365,107,231),(4,137,378,366,108,232),(5,138,379,367,109,233),(6,139,380,368,110,234),(7,140,371,369,101,235),(8,131,372,370,102,236),(9,132,373,361,103,237),(10,133,374,362,104,238),(11,166,254,305,476,342),(12,167,255,306,477,343),(13,168,256,307,478,344),(14,169,257,308,479,345),(15,170,258,309,480,346),(16,161,259,310,471,347),(17,162,260,301,472,348),(18,163,251,302,473,349),(19,164,252,303,474,350),(20,165,253,304,475,341),(21,152,324,178,43,461),(22,153,325,179,44,462),(23,154,326,180,45,463),(24,155,327,171,46,464),(25,156,328,172,47,465),(26,157,329,173,48,466),(27,158,330,174,49,467),(28,159,321,175,50,468),(29,160,322,176,41,469),(30,151,323,177,42,470),(31,358,340,142,459,449),(32,359,331,143,460,450),(33,360,332,144,451,441),(34,351,333,145,452,442),(35,352,334,146,453,443),(36,353,335,147,454,444),(37,354,336,148,455,445),(38,355,337,149,456,446),(39,356,338,150,457,447),(40,357,339,141,458,448),(51,74,192,190,282,120),(52,75,193,181,283,111),(53,76,194,182,284,112),(54,77,195,183,285,113),(55,78,196,184,286,114),(56,79,197,185,287,115),(57,80,198,186,288,116),(58,71,199,187,289,117),(59,72,200,188,290,118),(60,73,191,189,281,119),(61,390,400,229,271,299),(62,381,391,230,272,300),(63,382,392,221,273,291),(64,383,393,222,274,292),(65,384,394,223,275,293),(66,385,395,224,276,294),(67,386,396,225,277,295),(68,387,397,226,278,296),(69,388,398,227,279,297),(70,389,399,228,280,298),(81,270,208,218,407,99),(82,261,209,219,408,100),(83,262,210,220,409,91),(84,263,201,211,410,92),(85,264,202,212,401,93),(86,265,203,213,402,94),(87,266,204,214,403,95),(88,267,205,215,404,96),(89,268,206,216,405,97),(90,269,207,217,406,98),(121,426,438,320,241,412),(122,427,439,311,242,413),(123,428,440,312,243,414),(124,429,431,313,244,415),(125,430,432,314,245,416),(126,421,433,315,246,417),(127,422,434,316,247,418),(128,423,435,317,248,419),(129,424,436,318,249,420),(130,425,437,319,250,411)], [(1,14,363,308),(2,15,364,309),(3,16,365,310),(4,17,366,301),(5,18,367,302),(6,19,368,303),(7,20,369,304),(8,11,370,305),(9,12,361,306),(10,13,362,307),(21,392,178,291),(22,393,179,292),(23,394,180,293),(24,395,171,294),(25,396,172,295),(26,397,173,296),(27,398,174,297),(28,399,175,298),(29,400,176,299),(30,391,177,300),(31,261,142,408),(32,262,143,409),(33,263,144,410),(34,264,145,401),(35,265,146,402),(36,266,147,403),(37,267,148,404),(38,268,149,405),(39,269,150,406),(40,270,141,407),(41,271,160,390),(42,272,151,381),(43,273,152,382),(44,274,153,383),(45,275,154,384),(46,276,155,385),(47,277,156,386),(48,278,157,387),(49,279,158,388),(50,280,159,389),(51,430,190,245),(52,421,181,246),(53,422,182,247),(54,423,183,248),(55,424,184,249),(56,425,185,250),(57,426,186,241),(58,427,187,242),(59,428,188,243),(60,429,189,244),(61,469,229,322),(62,470,230,323),(63,461,221,324),(64,462,222,325),(65,463,223,326),(66,464,224,327),(67,465,225,328),(68,466,226,329),(69,467,227,330),(70,468,228,321),(71,122,289,311),(72,123,290,312),(73,124,281,313),(74,125,282,314),(75,126,283,315),(76,127,284,316),(77,128,285,317),(78,129,286,318),(79,130,287,319),(80,121,288,320),(81,458,218,357),(82,459,219,358),(83,460,220,359),(84,451,211,360),(85,452,212,351),(86,453,213,352),(87,454,214,353),(88,455,215,354),(89,456,216,355),(90,457,217,356),(91,450,210,331),(92,441,201,332),(93,442,202,333),(94,443,203,334),(95,444,204,335),(96,445,205,336),(97,446,206,337),(98,447,207,338),(99,448,208,339),(100,449,209,340),(101,253,140,341),(102,254,131,342),(103,255,132,343),(104,256,133,344),(105,257,134,345),(106,258,135,346),(107,259,136,347),(108,260,137,348),(109,251,138,349),(110,252,139,350),(111,433,193,417),(112,434,194,418),(113,435,195,419),(114,436,196,420),(115,437,197,411),(116,438,198,412),(117,439,199,413),(118,440,200,414),(119,431,191,415),(120,432,192,416),(161,377,471,231),(162,378,472,232),(163,379,473,233),(164,380,474,234),(165,371,475,235),(166,372,476,236),(167,373,477,237),(168,374,478,238),(169,375,479,239),(170,376,480,240)]])
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 | Dic3 | D6 | D6 | Dic6 | D12 | C5×S3 | C5×D4 | C5×Q8 | C5×Dic3 | S3×C10 | S3×C10 | C5×Dic6 | C5×D12 |
| kernel | C10×C4⋊Dic3 | C5×C4⋊Dic3 | Dic3×C2×C10 | C22×C60 | C2×C60 | C2×C4⋊Dic3 | C4⋊Dic3 | C22×Dic3 | C22×C12 | C2×C12 | C22×C20 | C2×C30 | C2×C30 | C2×C20 | C2×C20 | C22×C10 | C2×C10 | C2×C10 | C22×C4 | C2×C6 | C2×C6 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 4 | 16 | 8 | 4 | 32 | 1 | 2 | 2 | 4 | 2 | 1 | 4 | 4 | 4 | 8 | 8 | 16 | 8 | 4 | 16 | 16 |
Matrix representation of C10×C4⋊Dic3 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 60 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 38 | 46 |
| 0 | 0 | 15 | 23 |
| 1 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 36 | 51 |
| 0 | 0 | 26 | 25 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,58,0,0,0,0,58],[60,0,0,0,0,1,0,0,0,0,38,15,0,0,46,23],[1,0,0,0,0,60,0,0,0,0,0,60,0,0,1,1],[60,0,0,0,0,50,0,0,0,0,36,26,0,0,51,25] >;
C10×C4⋊Dic3 in GAP, Magma, Sage, TeX
C_{10}\times C_4\rtimes {\rm Dic}_3 % in TeX
G:=Group("C10xC4:Dic3"); // GroupNames label
G:=SmallGroup(480,804);
// by ID
G=gap.SmallGroup(480,804);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,1766,436,15686]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations