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G = C6×Dic20order 480 = 25·3·5

Direct product of C6 and Dic20

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

Aliases: C6×Dic20, C305Q16, C60.176D4, C24.74D10, C12.46D20, C120.87C22, C60.263C23, C51(C6×Q16), (C2×C40).6C6, C101(C3×Q16), C1511(C2×Q16), C4.8(C3×D20), C8.16(C6×D5), C40.18(C2×C6), C20.31(C3×D4), C2.14(C6×D20), (C2×C24).10D5, C6.83(C2×D20), C10.10(C6×D4), (C2×C6).55D20, (C2×C120).14C2, C30.284(C2×D4), (C2×C30).115D4, (C2×C12).432D10, C20.30(C22×C6), Dic10.7(C2×C6), (C2×Dic10).5C6, C22.14(C3×D20), (C2×C60).510C22, (C6×Dic10).16C2, C12.236(C22×D5), (C3×Dic10).49C22, C4.29(D5×C2×C6), (C2×C8).4(C3×D5), (C2×C4).83(C6×D5), (C2×C20).93(C2×C6), (C2×C10).19(C3×D4), SmallGroup(480,698)

Series: Derived Chief Lower central Upper central

C1C20 — C6×Dic20
C1C5C10C20C60C3×Dic10C6×Dic10 — C6×Dic20
C5C10C20 — C6×Dic20
C1C2×C6C2×C12C2×C24

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

Subgroups: 368 in 120 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×4], C22, C5, C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C10, C10 [×2], C12 [×2], C12 [×4], C2×C6, C15, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×4], C20 [×2], C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×6], C30, C30 [×2], C2×Q16, C40 [×2], Dic10 [×4], Dic10 [×2], C2×Dic5 [×2], C2×C20, C2×C24, C3×Q16 [×4], C6×Q8 [×2], C3×Dic5 [×4], C60 [×2], C2×C30, Dic20 [×4], C2×C40, C2×Dic10 [×2], C6×Q16, C120 [×2], C3×Dic10 [×4], C3×Dic10 [×2], C6×Dic5 [×2], C2×C60, C2×Dic20, C3×Dic20 [×4], C2×C120, C6×Dic10 [×2], C6×Dic20
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], Q16 [×2], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C2×Q16, D20 [×2], C22×D5, C3×Q16 [×2], C6×D4, C6×D5 [×3], Dic20 [×2], C2×D20, C6×Q16, C3×D20 [×2], D5×C2×C6, C2×Dic20, C3×Dic20 [×2], C6×D20, C6×Dic20

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

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

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

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

138 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F5A5B6A···6F8A8B8C8D10A···10F12A12B12C12D12E···12L15A15B15C15D20A···20H24A···24H30A···30L40A···40P60A···60P120A···120AF
order122233444444556···6888810···101212121212···121515151520···2024···2430···3040···4060···60120···120
size1111112220202020221···122222···2222220···2022222···22···22···22···22···22···2

138 irreducible representations

dim11111111222222222222222222
type+++++++-++++-
imageC1C2C2C2C3C6C6C6D4D4D5Q16D10D10C3×D4C3×D4C3×D5D20D20C3×Q16C6×D5C6×D5Dic20C3×D20C3×D20C3×Dic20
kernelC6×Dic20C3×Dic20C2×C120C6×Dic10C2×Dic20Dic20C2×C40C2×Dic10C60C2×C30C2×C24C30C24C2×C12C20C2×C10C2×C8C12C2×C6C10C8C2×C4C6C4C22C2
# reps1412282411244222444884168832

Matrix representation of C6×Dic20 in GL3(𝔽241) generated by

22600
0160
0016
,
100
0212221
02047
,
100
070149
0124171
G:=sub<GL(3,GF(241))| [226,0,0,0,16,0,0,0,16],[1,0,0,0,212,20,0,221,47],[1,0,0,0,70,124,0,149,171] >;

C6×Dic20 in GAP, Magma, Sage, TeX

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

G:=Group("C6xDic20");
// GroupNames label

G:=SmallGroup(480,698);
// by ID

G=gap.SmallGroup(480,698);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,590,394,2524,102,18822]);
// Polycyclic

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

׿
×
𝔽