Copied to
clipboard

G = C10×C4⋊Dic3order 480 = 25·3·5

Direct product of C10 and C4⋊Dic3

Series: Derived Chief Lower central Upper central

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

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 [×3], C2 [×4], C3, C4 [×4], C4 [×4], C22, C22 [×6], C5, C6 [×3], C6 [×4], C2×C4 [×6], C2×C4 [×8], C23, C10 [×3], C10 [×4], Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4, C22×C4 [×2], C20 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×6], C22×C6, C30 [×3], C30 [×4], C2×C4⋊C4, C2×C20 [×6], C2×C20 [×8], C22×C10, C4⋊Dic3 [×4], C22×Dic3 [×2], C22×C12, C5×Dic3 [×4], C60 [×4], C2×C30, C2×C30 [×6], C5×C4⋊C4 [×4], C22×C20, C22×C20 [×2], C2×C4⋊Dic3, C10×Dic3 [×4], C10×Dic3 [×4], C2×C60 [×6], C22×C30, C10×C4⋊C4, C5×C4⋊Dic3 [×4], Dic3×C2×C10 [×2], C22×C60, C10×C4⋊Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, C10 [×7], Dic3 [×4], D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C20 [×4], C2×C10 [×7], Dic6 [×2], D12 [×2], C2×Dic3 [×6], C22×S3, C5×S3, C2×C4⋊C4, C2×C20 [×6], C5×D4 [×2], C5×Q8 [×2], C22×C10, C4⋊Dic3 [×4], C2×Dic6, C2×D12, C22×Dic3, C5×Dic3 [×4], S3×C10 [×3], C5×C4⋊C4 [×4], C22×C20, D4×C10, Q8×C10, C2×C4⋊Dic3, C5×Dic6 [×2], C5×D12 [×2], C10×Dic3 [×6], S3×C2×C10, C10×C4⋊C4, C5×C4⋊Dic3 [×4], C10×Dic6, C10×D12, Dic3×C2×C10, C10×C4⋊Dic3

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

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

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

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

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

׿
×
𝔽