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## G = C6.Dic20order 480 = 25·3·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C6.Dic20
 Chief series C1 — C5 — C15 — C30 — C60 — C2×C60 — C6×Dic10 — C6.Dic20
 Lower central C15 — C30 — C60 — C6.Dic20
 Upper central C1 — C22 — C2×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, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, Q8, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, C20, C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8, C30, Q8⋊C4, C40, Dic10, Dic10, C2×Dic5, C2×C20, C2×C3⋊C8, C4⋊Dic3, C6×Q8, C3×Dic5, Dic15, C60, C2×C30, C4⋊Dic5, C2×C40, C2×Dic10, Q82Dic3, C5×C3⋊C8, C3×Dic10, C3×Dic10, C6×Dic5, C2×Dic15, C2×C60, C20.44D4, C10×C3⋊C8, C605C4, C6×Dic10, C6.Dic20
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, SD16, Q16, D10, C2×Dic3, C3⋊D4, 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
(1 323 230 134 365 164)(2 165 366 135 231 324)(3 325 232 136 367 166)(4 167 368 137 233 326)(5 327 234 138 369 168)(6 169 370 139 235 328)(7 329 236 140 371 170)(8 171 372 141 237 330)(9 331 238 142 373 172)(10 173 374 143 239 332)(11 333 240 144 375 174)(12 175 376 145 201 334)(13 335 202 146 377 176)(14 177 378 147 203 336)(15 337 204 148 379 178)(16 179 380 149 205 338)(17 339 206 150 381 180)(18 181 382 151 207 340)(19 341 208 152 383 182)(20 183 384 153 209 342)(21 343 210 154 385 184)(22 185 386 155 211 344)(23 345 212 156 387 186)(24 187 388 157 213 346)(25 347 214 158 389 188)(26 189 390 159 215 348)(27 349 216 160 391 190)(28 191 392 121 217 350)(29 351 218 122 393 192)(30 193 394 123 219 352)(31 353 220 124 395 194)(32 195 396 125 221 354)(33 355 222 126 397 196)(34 197 398 127 223 356)(35 357 224 128 399 198)(36 199 400 129 225 358)(37 359 226 130 361 200)(38 161 362 131 227 360)(39 321 228 132 363 162)(40 163 364 133 229 322)(41 450 264 311 431 87)(42 88 432 312 265 451)(43 452 266 313 433 89)(44 90 434 314 267 453)(45 454 268 315 435 91)(46 92 436 316 269 455)(47 456 270 317 437 93)(48 94 438 318 271 457)(49 458 272 319 439 95)(50 96 440 320 273 459)(51 460 274 281 401 97)(52 98 402 282 275 461)(53 462 276 283 403 99)(54 100 404 284 277 463)(55 464 278 285 405 101)(56 102 406 286 279 465)(57 466 280 287 407 103)(58 104 408 288 241 467)(59 468 242 289 409 105)(60 106 410 290 243 469)(61 470 244 291 411 107)(62 108 412 292 245 471)(63 472 246 293 413 109)(64 110 414 294 247 473)(65 474 248 295 415 111)(66 112 416 296 249 475)(67 476 250 297 417 113)(68 114 418 298 251 477)(69 478 252 299 419 115)(70 116 420 300 253 479)(71 480 254 301 421 117)(72 118 422 302 255 441)(73 442 256 303 423 119)(74 120 424 304 257 443)(75 444 258 305 425 81)(76 82 426 306 259 445)(77 446 260 307 427 83)(78 84 428 308 261 447)(79 448 262 309 429 85)(80 86 430 310 263 449)
(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)
(1 300 21 320)(2 69 22 49)(3 298 23 318)(4 67 24 47)(5 296 25 316)(6 65 26 45)(7 294 27 314)(8 63 28 43)(9 292 29 312)(10 61 30 41)(11 290 31 310)(12 59 32 79)(13 288 33 308)(14 57 34 77)(15 286 35 306)(16 55 36 75)(17 284 37 304)(18 53 38 73)(19 282 39 302)(20 51 40 71)(42 142 62 122)(44 140 64 160)(46 138 66 158)(48 136 68 156)(50 134 70 154)(52 132 72 152)(54 130 74 150)(56 128 76 148)(58 126 78 146)(60 124 80 144)(81 338 101 358)(82 379 102 399)(83 336 103 356)(84 377 104 397)(85 334 105 354)(86 375 106 395)(87 332 107 352)(88 373 108 393)(89 330 109 350)(90 371 110 391)(91 328 111 348)(92 369 112 389)(93 326 113 346)(94 367 114 387)(95 324 115 344)(96 365 116 385)(97 322 117 342)(98 363 118 383)(99 360 119 340)(100 361 120 381)(121 313 141 293)(123 311 143 291)(125 309 145 289)(127 307 147 287)(129 305 149 285)(131 303 151 283)(133 301 153 281)(135 299 155 319)(137 297 157 317)(139 295 159 315)(161 442 181 462)(162 422 182 402)(163 480 183 460)(164 420 184 440)(165 478 185 458)(166 418 186 438)(167 476 187 456)(168 416 188 436)(169 474 189 454)(170 414 190 434)(171 472 191 452)(172 412 192 432)(173 470 193 450)(174 410 194 430)(175 468 195 448)(176 408 196 428)(177 466 197 446)(178 406 198 426)(179 464 199 444)(180 404 200 424)(201 409 221 429)(202 467 222 447)(203 407 223 427)(204 465 224 445)(205 405 225 425)(206 463 226 443)(207 403 227 423)(208 461 228 441)(209 401 229 421)(210 459 230 479)(211 439 231 419)(212 457 232 477)(213 437 233 417)(214 455 234 475)(215 435 235 415)(216 453 236 473)(217 433 237 413)(218 451 238 471)(219 431 239 411)(220 449 240 469)(241 355 261 335)(242 396 262 376)(243 353 263 333)(244 394 264 374)(245 351 265 331)(246 392 266 372)(247 349 267 329)(248 390 268 370)(249 347 269 327)(250 388 270 368)(251 345 271 325)(252 386 272 366)(253 343 273 323)(254 384 274 364)(255 341 275 321)(256 382 276 362)(257 339 277 359)(258 380 278 400)(259 337 279 357)(260 378 280 398)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A ··· 20H 30A ··· 30F 40A ··· 40P 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 12 12 15 15 20 ··· 20 30 ··· 30 40 ··· 40 60 ··· 60 size 1 1 1 1 2 2 2 20 20 60 60 2 2 2 2 2 6 6 6 6 2 ··· 2 4 4 20 20 20 20 4 4 2 ··· 2 4 ··· 4 6 ··· 6 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + - + - + + - + - + - - + + - image C1 C2 C2 C2 C4 S3 D4 D4 D5 Dic3 D6 SD16 Q16 D10 C3⋊D4 C3⋊D4 C4×D5 C5⋊D4 D20 C40⋊C2 Dic20 Q8⋊2S3 C3⋊Q16 S3×D5 D5×Dic3 C15⋊D4 C3⋊D20 C15⋊SD16 C3⋊Dic20 kernel C6.Dic20 C10×C3⋊C8 C60⋊5C4 C6×Dic10 C3×Dic10 C2×Dic10 C60 C2×C30 C2×C3⋊C8 Dic10 C2×C20 C30 C30 C2×C12 C20 C2×C10 C12 C12 C2×C6 C6 C6 C10 C10 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 2 2 1 2 2 2 2 2 4 4 4 8 8 1 1 2 2 2 2 4 4

Matrix representation of C6.Dic20 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 240 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 192 165 0 0 0 0 76 172 0 0 0 0 0 0 240 0 0 0 0 0 1 1 0 0 0 0 0 0 169 37 0 0 0 0 204 173
,
 65 34 0 0 0 0 216 176 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 64 0 0 0 0 0 46 177

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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