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G = C6.Dic20order 480 = 25·3·5

1st non-split extension by C6 of Dic20 acting via Dic20/Dic10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.34D4, C30.2Q16, C6.3Dic20, C30.8SD16, Dic104Dic3, C12.5(C4×D5), C60.92(C2×C4), (C2×C30).18D4, (C2×C6).31D20, C4.2(D5×Dic3), (C3×Dic10)⋊7C4, C154(Q8⋊C4), (C2×C20).281D6, (C2×C12).52D10, C6.5(C40⋊C2), C52(Q82Dic3), C605C4.15C2, C33(C20.44D4), C20.80(C3⋊D4), C4.22(C15⋊D4), C12.14(C5⋊D4), (C2×Dic10).1S3, (C6×Dic10).2C2, C20.38(C2×Dic3), C2.1(C3⋊Dic20), C10.1(C3⋊Q16), C30.54(C22⋊C4), (C2×C60).100C22, C10.2(Q82S3), C2.2(C15⋊SD16), C6.29(D10⋊C4), C2.8(D10⋊Dic3), C22.14(C3⋊D20), C10.18(C6.D4), (C2×C3⋊C8).1D5, (C10×C3⋊C8).2C2, (C2×C4).84(S3×D5), (C2×C10).25(C3⋊D4), SmallGroup(480,47)

Series: Derived Chief Lower central Upper central

C1C60 — C6.Dic20
C1C5C15C30C60C2×C60C6×Dic10 — C6.Dic20
C15C30C60 — C6.Dic20
C1C22C2×C4

Generators and relations for C6.Dic20
 G = < a,b,c | a6=b40=1, c2=b20, bab-1=a-1, ac=ca, cbc-1=a3b-1 >

Subgroups: 380 in 84 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×3], C10 [×3], Dic3, C12 [×2], C12 [×2], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5 [×3], C20 [×2], C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8 [×3], C30 [×3], Q8⋊C4, C40, Dic10 [×2], Dic10, C2×Dic5 [×2], C2×C20, C2×C3⋊C8, C4⋊Dic3, C6×Q8, C3×Dic5 [×2], Dic15, C60 [×2], C2×C30, C4⋊Dic5, C2×C40, C2×Dic10, Q82Dic3, C5×C3⋊C8, C3×Dic10 [×2], C3×Dic10, C6×Dic5, C2×Dic15, C2×C60, C20.44D4, C10×C3⋊C8, C605C4, C6×Dic10, C6.Dic20
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, SD16, Q16, D10, C2×Dic3, C3⋊D4 [×2], Q8⋊C4, C4×D5, D20, C5⋊D4, Q82S3, C3⋊Q16, C6.D4, S3×D5, C40⋊C2, Dic20, D10⋊C4, Q82Dic3, D5×Dic3, C15⋊D4, C3⋊D20, C20.44D4, C15⋊SD16, C3⋊Dic20, D10⋊Dic3, C6.Dic20

Smallest permutation representation of C6.Dic20
Regular action on 480 points
Generators in S480
(1 88 398 354 302 265)(2 266 303 355 399 89)(3 90 400 356 304 267)(4 268 305 357 361 91)(5 92 362 358 306 269)(6 270 307 359 363 93)(7 94 364 360 308 271)(8 272 309 321 365 95)(9 96 366 322 310 273)(10 274 311 323 367 97)(11 98 368 324 312 275)(12 276 313 325 369 99)(13 100 370 326 314 277)(14 278 315 327 371 101)(15 102 372 328 316 279)(16 280 317 329 373 103)(17 104 374 330 318 241)(18 242 319 331 375 105)(19 106 376 332 320 243)(20 244 281 333 377 107)(21 108 378 334 282 245)(22 246 283 335 379 109)(23 110 380 336 284 247)(24 248 285 337 381 111)(25 112 382 338 286 249)(26 250 287 339 383 113)(27 114 384 340 288 251)(28 252 289 341 385 115)(29 116 386 342 290 253)(30 254 291 343 387 117)(31 118 388 344 292 255)(32 256 293 345 389 119)(33 120 390 346 294 257)(34 258 295 347 391 81)(35 82 392 348 296 259)(36 260 297 349 393 83)(37 84 394 350 298 261)(38 262 299 351 395 85)(39 86 396 352 300 263)(40 264 301 353 397 87)(41 153 455 235 183 437)(42 438 184 236 456 154)(43 155 457 237 185 439)(44 440 186 238 458 156)(45 157 459 239 187 401)(46 402 188 240 460 158)(47 159 461 201 189 403)(48 404 190 202 462 160)(49 121 463 203 191 405)(50 406 192 204 464 122)(51 123 465 205 193 407)(52 408 194 206 466 124)(53 125 467 207 195 409)(54 410 196 208 468 126)(55 127 469 209 197 411)(56 412 198 210 470 128)(57 129 471 211 199 413)(58 414 200 212 472 130)(59 131 473 213 161 415)(60 416 162 214 474 132)(61 133 475 215 163 417)(62 418 164 216 476 134)(63 135 477 217 165 419)(64 420 166 218 478 136)(65 137 479 219 167 421)(66 422 168 220 480 138)(67 139 441 221 169 423)(68 424 170 222 442 140)(69 141 443 223 171 425)(70 426 172 224 444 142)(71 143 445 225 173 427)(72 428 174 226 446 144)(73 145 447 227 175 429)(74 430 176 228 448 146)(75 147 449 229 177 431)(76 432 178 230 450 148)(77 149 451 231 179 433)(78 434 180 232 452 150)(79 151 453 233 181 435)(80 436 182 234 454 152)
(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 42 21 62)(2 235 22 215)(3 80 23 60)(4 233 24 213)(5 78 25 58)(6 231 26 211)(7 76 27 56)(8 229 28 209)(9 74 29 54)(10 227 30 207)(11 72 31 52)(12 225 32 205)(13 70 33 50)(14 223 34 203)(15 68 35 48)(16 221 36 201)(17 66 37 46)(18 219 38 239)(19 64 39 44)(20 217 40 237)(41 335 61 355)(43 333 63 353)(45 331 65 351)(47 329 67 349)(49 327 69 347)(51 325 71 345)(53 323 73 343)(55 321 75 341)(57 359 77 339)(59 357 79 337)(81 463 101 443)(82 404 102 424)(83 461 103 441)(84 402 104 422)(85 459 105 479)(86 440 106 420)(87 457 107 477)(88 438 108 418)(89 455 109 475)(90 436 110 416)(91 453 111 473)(92 434 112 414)(93 451 113 471)(94 432 114 412)(95 449 115 469)(96 430 116 410)(97 447 117 467)(98 428 118 408)(99 445 119 465)(100 426 120 406)(121 371 141 391)(122 277 142 257)(123 369 143 389)(124 275 144 255)(125 367 145 387)(126 273 146 253)(127 365 147 385)(128 271 148 251)(129 363 149 383)(130 269 150 249)(131 361 151 381)(132 267 152 247)(133 399 153 379)(134 265 154 245)(135 397 155 377)(136 263 156 243)(137 395 157 375)(138 261 158 241)(139 393 159 373)(140 259 160 279)(161 268 181 248)(162 400 182 380)(163 266 183 246)(164 398 184 378)(165 264 185 244)(166 396 186 376)(167 262 187 242)(168 394 188 374)(169 260 189 280)(170 392 190 372)(171 258 191 278)(172 390 192 370)(173 256 193 276)(174 388 194 368)(175 254 195 274)(176 386 196 366)(177 252 197 272)(178 384 198 364)(179 250 199 270)(180 382 200 362)(202 328 222 348)(204 326 224 346)(206 324 226 344)(208 322 228 342)(210 360 230 340)(212 358 232 338)(214 356 234 336)(216 354 236 334)(218 352 238 332)(220 350 240 330)(281 419 301 439)(282 476 302 456)(283 417 303 437)(284 474 304 454)(285 415 305 435)(286 472 306 452)(287 413 307 433)(288 470 308 450)(289 411 309 431)(290 468 310 448)(291 409 311 429)(292 466 312 446)(293 407 313 427)(294 464 314 444)(295 405 315 425)(296 462 316 442)(297 403 317 423)(298 460 318 480)(299 401 319 421)(300 458 320 478)

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

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

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

72 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D12E12F15A15B20A···20H30A···30F40A···40P60A···60H
order1222344444455666888810···10121212121212151520···2030···3040···4060···60
size1111222202060602222266662···24420202020442···24···46···64···4

72 irreducible representations

dim11111222222222222222244444444
type++++++++-+-++-+-+--++-
imageC1C2C2C2C4S3D4D4D5Dic3D6SD16Q16D10C3⋊D4C3⋊D4C4×D5C5⋊D4D20C40⋊C2Dic20Q82S3C3⋊Q16S3×D5D5×Dic3C15⋊D4C3⋊D20C15⋊SD16C3⋊Dic20
kernelC6.Dic20C10×C3⋊C8C605C4C6×Dic10C3×Dic10C2×Dic10C60C2×C30C2×C3⋊C8Dic10C2×C20C30C30C2×C12C20C2×C10C12C12C2×C6C6C6C10C10C2×C4C4C4C22C2C2
# reps11114111221222224448811222244

Matrix representation of C6.Dic20 in GL6(𝔽241)

24000000
02400000
0024024000
001000
00002400
00000240
,
1921650000
761720000
00240000
001100
000016937
0000204173
,
65340000
2161760000
00240000
00024000
0000640
000046177

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,1,0,0,0,0,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[192,76,0,0,0,0,165,172,0,0,0,0,0,0,240,1,0,0,0,0,0,1,0,0,0,0,0,0,169,204,0,0,0,0,37,173],[65,216,0,0,0,0,34,176,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,46,0,0,0,0,0,177] >;

C6.Dic20 in GAP, Magma, Sage, TeX

C_6.{\rm Dic}_{20}
% in TeX

G:=Group("C6.Dic20");
// GroupNames label

G:=SmallGroup(480,47);
// by ID

G=gap.SmallGroup(480,47);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,85,92,422,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^6=b^40=1,c^2=b^20,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b^-1>;
// generators/relations

׿
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