direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6×Dic20, C30⋊5Q16, C60.176D4, C24.74D10, C12.46D20, C120.87C22, C60.263C23, C5⋊1(C6×Q16), (C2×C40).6C6, C10⋊1(C3×Q16), C15⋊11(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
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, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, C12, C12, C2×C6, C15, C2×C8, Q16, C2×Q8, Dic5, C20, C2×C10, C24, C2×C12, C2×C12, C3×Q8, C30, C30, C2×Q16, C40, Dic10, Dic10, C2×Dic5, C2×C20, C2×C24, C3×Q16, C6×Q8, C3×Dic5, C60, C2×C30, Dic20, C2×C40, C2×Dic10, C6×Q16, C120, C3×Dic10, C3×Dic10, C6×Dic5, C2×C60, C2×Dic20, C3×Dic20, C2×C120, C6×Dic10, C6×Dic20
Quotients: C1, C2, C3, C22, C6, D4, C23, D5, C2×C6, Q16, C2×D4, D10, C3×D4, C22×C6, C3×D5, C2×Q16, D20, C22×D5, C3×Q16, C6×D4, C6×D5, Dic20, C2×D20, C6×Q16, C3×D20, D5×C2×C6, C2×Dic20, C3×Dic20, C6×D20, C6×Dic20
►Regular action on 480 points
Generators in S480
(1 88 189 202 370 267)(2 89 190 203 371 268)(3 90 191 204 372 269)(4 91 192 205 373 270)(5 92 193 206 374 271)(6 93 194 207 375 272)(7 94 195 208 376 273)(8 95 196 209 377 274)(9 96 197 210 378 275)(10 97 198 211 379 276)(11 98 199 212 380 277)(12 99 200 213 381 278)(13 100 161 214 382 279)(14 101 162 215 383 280)(15 102 163 216 384 241)(16 103 164 217 385 242)(17 104 165 218 386 243)(18 105 166 219 387 244)(19 106 167 220 388 245)(20 107 168 221 389 246)(21 108 169 222 390 247)(22 109 170 223 391 248)(23 110 171 224 392 249)(24 111 172 225 393 250)(25 112 173 226 394 251)(26 113 174 227 395 252)(27 114 175 228 396 253)(28 115 176 229 397 254)(29 116 177 230 398 255)(30 117 178 231 399 256)(31 118 179 232 400 257)(32 119 180 233 361 258)(33 120 181 234 362 259)(34 81 182 235 363 260)(35 82 183 236 364 261)(36 83 184 237 365 262)(37 84 185 238 366 263)(38 85 186 239 367 264)(39 86 187 240 368 265)(40 87 188 201 369 266)(41 300 429 156 470 329)(42 301 430 157 471 330)(43 302 431 158 472 331)(44 303 432 159 473 332)(45 304 433 160 474 333)(46 305 434 121 475 334)(47 306 435 122 476 335)(48 307 436 123 477 336)(49 308 437 124 478 337)(50 309 438 125 479 338)(51 310 439 126 480 339)(52 311 440 127 441 340)(53 312 401 128 442 341)(54 313 402 129 443 342)(55 314 403 130 444 343)(56 315 404 131 445 344)(57 316 405 132 446 345)(58 317 406 133 447 346)(59 318 407 134 448 347)(60 319 408 135 449 348)(61 320 409 136 450 349)(62 281 410 137 451 350)(63 282 411 138 452 351)(64 283 412 139 453 352)(65 284 413 140 454 353)(66 285 414 141 455 354)(67 286 415 142 456 355)(68 287 416 143 457 356)(69 288 417 144 458 357)(70 289 418 145 459 358)(71 290 419 146 460 359)(72 291 420 147 461 360)(73 292 421 148 462 321)(74 293 422 149 463 322)(75 294 423 150 464 323)(76 295 424 151 465 324)(77 296 425 152 466 325)(78 297 426 153 467 326)(79 298 427 154 468 327)(80 299 428 155 469 328)
(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)
(1 470 21 450)(2 469 22 449)(3 468 23 448)(4 467 24 447)(5 466 25 446)(6 465 26 445)(7 464 27 444)(8 463 28 443)(9 462 29 442)(10 461 30 441)(11 460 31 480)(12 459 32 479)(13 458 33 478)(14 457 34 477)(15 456 35 476)(16 455 36 475)(17 454 37 474)(18 453 38 473)(19 452 39 472)(20 451 40 471)(41 169 61 189)(42 168 62 188)(43 167 63 187)(44 166 64 186)(45 165 65 185)(46 164 66 184)(47 163 67 183)(48 162 68 182)(49 161 69 181)(50 200 70 180)(51 199 71 179)(52 198 72 178)(53 197 73 177)(54 196 74 176)(55 195 75 175)(56 194 76 174)(57 193 77 173)(58 192 78 172)(59 191 79 171)(60 190 80 170)(81 336 101 356)(82 335 102 355)(83 334 103 354)(84 333 104 353)(85 332 105 352)(86 331 106 351)(87 330 107 350)(88 329 108 349)(89 328 109 348)(90 327 110 347)(91 326 111 346)(92 325 112 345)(93 324 113 344)(94 323 114 343)(95 322 115 342)(96 321 116 341)(97 360 117 340)(98 359 118 339)(99 358 119 338)(100 357 120 337)(121 242 141 262)(122 241 142 261)(123 280 143 260)(124 279 144 259)(125 278 145 258)(126 277 146 257)(127 276 147 256)(128 275 148 255)(129 274 149 254)(130 273 150 253)(131 272 151 252)(132 271 152 251)(133 270 153 250)(134 269 154 249)(135 268 155 248)(136 267 156 247)(137 266 157 246)(138 265 158 245)(139 264 159 244)(140 263 160 243)(201 301 221 281)(202 300 222 320)(203 299 223 319)(204 298 224 318)(205 297 225 317)(206 296 226 316)(207 295 227 315)(208 294 228 314)(209 293 229 313)(210 292 230 312)(211 291 231 311)(212 290 232 310)(213 289 233 309)(214 288 234 308)(215 287 235 307)(216 286 236 306)(217 285 237 305)(218 284 238 304)(219 283 239 303)(220 282 240 302)(361 438 381 418)(362 437 382 417)(363 436 383 416)(364 435 384 415)(365 434 385 414)(366 433 386 413)(367 432 387 412)(368 431 388 411)(369 430 389 410)(370 429 390 409)(371 428 391 408)(372 427 392 407)(373 426 393 406)(374 425 394 405)(375 424 395 404)(376 423 396 403)(377 422 397 402)(378 421 398 401)(379 420 399 440)(380 419 400 439)
G:=sub<Sym(480)| (1,88,189,202,370,267)(2,89,190,203,371,268)(3,90,191,204,372,269)(4,91,192,205,373,270)(5,92,193,206,374,271)(6,93,194,207,375,272)(7,94,195,208,376,273)(8,95,196,209,377,274)(9,96,197,210,378,275)(10,97,198,211,379,276)(11,98,199,212,380,277)(12,99,200,213,381,278)(13,100,161,214,382,279)(14,101,162,215,383,280)(15,102,163,216,384,241)(16,103,164,217,385,242)(17,104,165,218,386,243)(18,105,166,219,387,244)(19,106,167,220,388,245)(20,107,168,221,389,246)(21,108,169,222,390,247)(22,109,170,223,391,248)(23,110,171,224,392,249)(24,111,172,225,393,250)(25,112,173,226,394,251)(26,113,174,227,395,252)(27,114,175,228,396,253)(28,115,176,229,397,254)(29,116,177,230,398,255)(30,117,178,231,399,256)(31,118,179,232,400,257)(32,119,180,233,361,258)(33,120,181,234,362,259)(34,81,182,235,363,260)(35,82,183,236,364,261)(36,83,184,237,365,262)(37,84,185,238,366,263)(38,85,186,239,367,264)(39,86,187,240,368,265)(40,87,188,201,369,266)(41,300,429,156,470,329)(42,301,430,157,471,330)(43,302,431,158,472,331)(44,303,432,159,473,332)(45,304,433,160,474,333)(46,305,434,121,475,334)(47,306,435,122,476,335)(48,307,436,123,477,336)(49,308,437,124,478,337)(50,309,438,125,479,338)(51,310,439,126,480,339)(52,311,440,127,441,340)(53,312,401,128,442,341)(54,313,402,129,443,342)(55,314,403,130,444,343)(56,315,404,131,445,344)(57,316,405,132,446,345)(58,317,406,133,447,346)(59,318,407,134,448,347)(60,319,408,135,449,348)(61,320,409,136,450,349)(62,281,410,137,451,350)(63,282,411,138,452,351)(64,283,412,139,453,352)(65,284,413,140,454,353)(66,285,414,141,455,354)(67,286,415,142,456,355)(68,287,416,143,457,356)(69,288,417,144,458,357)(70,289,418,145,459,358)(71,290,419,146,460,359)(72,291,420,147,461,360)(73,292,421,148,462,321)(74,293,422,149,463,322)(75,294,423,150,464,323)(76,295,424,151,465,324)(77,296,425,152,466,325)(78,297,426,153,467,326)(79,298,427,154,468,327)(80,299,428,155,469,328), (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), (1,470,21,450)(2,469,22,449)(3,468,23,448)(4,467,24,447)(5,466,25,446)(6,465,26,445)(7,464,27,444)(8,463,28,443)(9,462,29,442)(10,461,30,441)(11,460,31,480)(12,459,32,479)(13,458,33,478)(14,457,34,477)(15,456,35,476)(16,455,36,475)(17,454,37,474)(18,453,38,473)(19,452,39,472)(20,451,40,471)(41,169,61,189)(42,168,62,188)(43,167,63,187)(44,166,64,186)(45,165,65,185)(46,164,66,184)(47,163,67,183)(48,162,68,182)(49,161,69,181)(50,200,70,180)(51,199,71,179)(52,198,72,178)(53,197,73,177)(54,196,74,176)(55,195,75,175)(56,194,76,174)(57,193,77,173)(58,192,78,172)(59,191,79,171)(60,190,80,170)(81,336,101,356)(82,335,102,355)(83,334,103,354)(84,333,104,353)(85,332,105,352)(86,331,106,351)(87,330,107,350)(88,329,108,349)(89,328,109,348)(90,327,110,347)(91,326,111,346)(92,325,112,345)(93,324,113,344)(94,323,114,343)(95,322,115,342)(96,321,116,341)(97,360,117,340)(98,359,118,339)(99,358,119,338)(100,357,120,337)(121,242,141,262)(122,241,142,261)(123,280,143,260)(124,279,144,259)(125,278,145,258)(126,277,146,257)(127,276,147,256)(128,275,148,255)(129,274,149,254)(130,273,150,253)(131,272,151,252)(132,271,152,251)(133,270,153,250)(134,269,154,249)(135,268,155,248)(136,267,156,247)(137,266,157,246)(138,265,158,245)(139,264,159,244)(140,263,160,243)(201,301,221,281)(202,300,222,320)(203,299,223,319)(204,298,224,318)(205,297,225,317)(206,296,226,316)(207,295,227,315)(208,294,228,314)(209,293,229,313)(210,292,230,312)(211,291,231,311)(212,290,232,310)(213,289,233,309)(214,288,234,308)(215,287,235,307)(216,286,236,306)(217,285,237,305)(218,284,238,304)(219,283,239,303)(220,282,240,302)(361,438,381,418)(362,437,382,417)(363,436,383,416)(364,435,384,415)(365,434,385,414)(366,433,386,413)(367,432,387,412)(368,431,388,411)(369,430,389,410)(370,429,390,409)(371,428,391,408)(372,427,392,407)(373,426,393,406)(374,425,394,405)(375,424,395,404)(376,423,396,403)(377,422,397,402)(378,421,398,401)(379,420,399,440)(380,419,400,439)>;
G:=Group( (1,88,189,202,370,267)(2,89,190,203,371,268)(3,90,191,204,372,269)(4,91,192,205,373,270)(5,92,193,206,374,271)(6,93,194,207,375,272)(7,94,195,208,376,273)(8,95,196,209,377,274)(9,96,197,210,378,275)(10,97,198,211,379,276)(11,98,199,212,380,277)(12,99,200,213,381,278)(13,100,161,214,382,279)(14,101,162,215,383,280)(15,102,163,216,384,241)(16,103,164,217,385,242)(17,104,165,218,386,243)(18,105,166,219,387,244)(19,106,167,220,388,245)(20,107,168,221,389,246)(21,108,169,222,390,247)(22,109,170,223,391,248)(23,110,171,224,392,249)(24,111,172,225,393,250)(25,112,173,226,394,251)(26,113,174,227,395,252)(27,114,175,228,396,253)(28,115,176,229,397,254)(29,116,177,230,398,255)(30,117,178,231,399,256)(31,118,179,232,400,257)(32,119,180,233,361,258)(33,120,181,234,362,259)(34,81,182,235,363,260)(35,82,183,236,364,261)(36,83,184,237,365,262)(37,84,185,238,366,263)(38,85,186,239,367,264)(39,86,187,240,368,265)(40,87,188,201,369,266)(41,300,429,156,470,329)(42,301,430,157,471,330)(43,302,431,158,472,331)(44,303,432,159,473,332)(45,304,433,160,474,333)(46,305,434,121,475,334)(47,306,435,122,476,335)(48,307,436,123,477,336)(49,308,437,124,478,337)(50,309,438,125,479,338)(51,310,439,126,480,339)(52,311,440,127,441,340)(53,312,401,128,442,341)(54,313,402,129,443,342)(55,314,403,130,444,343)(56,315,404,131,445,344)(57,316,405,132,446,345)(58,317,406,133,447,346)(59,318,407,134,448,347)(60,319,408,135,449,348)(61,320,409,136,450,349)(62,281,410,137,451,350)(63,282,411,138,452,351)(64,283,412,139,453,352)(65,284,413,140,454,353)(66,285,414,141,455,354)(67,286,415,142,456,355)(68,287,416,143,457,356)(69,288,417,144,458,357)(70,289,418,145,459,358)(71,290,419,146,460,359)(72,291,420,147,461,360)(73,292,421,148,462,321)(74,293,422,149,463,322)(75,294,423,150,464,323)(76,295,424,151,465,324)(77,296,425,152,466,325)(78,297,426,153,467,326)(79,298,427,154,468,327)(80,299,428,155,469,328), (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), (1,470,21,450)(2,469,22,449)(3,468,23,448)(4,467,24,447)(5,466,25,446)(6,465,26,445)(7,464,27,444)(8,463,28,443)(9,462,29,442)(10,461,30,441)(11,460,31,480)(12,459,32,479)(13,458,33,478)(14,457,34,477)(15,456,35,476)(16,455,36,475)(17,454,37,474)(18,453,38,473)(19,452,39,472)(20,451,40,471)(41,169,61,189)(42,168,62,188)(43,167,63,187)(44,166,64,186)(45,165,65,185)(46,164,66,184)(47,163,67,183)(48,162,68,182)(49,161,69,181)(50,200,70,180)(51,199,71,179)(52,198,72,178)(53,197,73,177)(54,196,74,176)(55,195,75,175)(56,194,76,174)(57,193,77,173)(58,192,78,172)(59,191,79,171)(60,190,80,170)(81,336,101,356)(82,335,102,355)(83,334,103,354)(84,333,104,353)(85,332,105,352)(86,331,106,351)(87,330,107,350)(88,329,108,349)(89,328,109,348)(90,327,110,347)(91,326,111,346)(92,325,112,345)(93,324,113,344)(94,323,114,343)(95,322,115,342)(96,321,116,341)(97,360,117,340)(98,359,118,339)(99,358,119,338)(100,357,120,337)(121,242,141,262)(122,241,142,261)(123,280,143,260)(124,279,144,259)(125,278,145,258)(126,277,146,257)(127,276,147,256)(128,275,148,255)(129,274,149,254)(130,273,150,253)(131,272,151,252)(132,271,152,251)(133,270,153,250)(134,269,154,249)(135,268,155,248)(136,267,156,247)(137,266,157,246)(138,265,158,245)(139,264,159,244)(140,263,160,243)(201,301,221,281)(202,300,222,320)(203,299,223,319)(204,298,224,318)(205,297,225,317)(206,296,226,316)(207,295,227,315)(208,294,228,314)(209,293,229,313)(210,292,230,312)(211,291,231,311)(212,290,232,310)(213,289,233,309)(214,288,234,308)(215,287,235,307)(216,286,236,306)(217,285,237,305)(218,284,238,304)(219,283,239,303)(220,282,240,302)(361,438,381,418)(362,437,382,417)(363,436,383,416)(364,435,384,415)(365,434,385,414)(366,433,386,413)(367,432,387,412)(368,431,388,411)(369,430,389,410)(370,429,390,409)(371,428,391,408)(372,427,392,407)(373,426,393,406)(374,425,394,405)(375,424,395,404)(376,423,396,403)(377,422,397,402)(378,421,398,401)(379,420,399,440)(380,419,400,439) );
G=PermutationGroup([[(1,88,189,202,370,267),(2,89,190,203,371,268),(3,90,191,204,372,269),(4,91,192,205,373,270),(5,92,193,206,374,271),(6,93,194,207,375,272),(7,94,195,208,376,273),(8,95,196,209,377,274),(9,96,197,210,378,275),(10,97,198,211,379,276),(11,98,199,212,380,277),(12,99,200,213,381,278),(13,100,161,214,382,279),(14,101,162,215,383,280),(15,102,163,216,384,241),(16,103,164,217,385,242),(17,104,165,218,386,243),(18,105,166,219,387,244),(19,106,167,220,388,245),(20,107,168,221,389,246),(21,108,169,222,390,247),(22,109,170,223,391,248),(23,110,171,224,392,249),(24,111,172,225,393,250),(25,112,173,226,394,251),(26,113,174,227,395,252),(27,114,175,228,396,253),(28,115,176,229,397,254),(29,116,177,230,398,255),(30,117,178,231,399,256),(31,118,179,232,400,257),(32,119,180,233,361,258),(33,120,181,234,362,259),(34,81,182,235,363,260),(35,82,183,236,364,261),(36,83,184,237,365,262),(37,84,185,238,366,263),(38,85,186,239,367,264),(39,86,187,240,368,265),(40,87,188,201,369,266),(41,300,429,156,470,329),(42,301,430,157,471,330),(43,302,431,158,472,331),(44,303,432,159,473,332),(45,304,433,160,474,333),(46,305,434,121,475,334),(47,306,435,122,476,335),(48,307,436,123,477,336),(49,308,437,124,478,337),(50,309,438,125,479,338),(51,310,439,126,480,339),(52,311,440,127,441,340),(53,312,401,128,442,341),(54,313,402,129,443,342),(55,314,403,130,444,343),(56,315,404,131,445,344),(57,316,405,132,446,345),(58,317,406,133,447,346),(59,318,407,134,448,347),(60,319,408,135,449,348),(61,320,409,136,450,349),(62,281,410,137,451,350),(63,282,411,138,452,351),(64,283,412,139,453,352),(65,284,413,140,454,353),(66,285,414,141,455,354),(67,286,415,142,456,355),(68,287,416,143,457,356),(69,288,417,144,458,357),(70,289,418,145,459,358),(71,290,419,146,460,359),(72,291,420,147,461,360),(73,292,421,148,462,321),(74,293,422,149,463,322),(75,294,423,150,464,323),(76,295,424,151,465,324),(77,296,425,152,466,325),(78,297,426,153,467,326),(79,298,427,154,468,327),(80,299,428,155,469,328)], [(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)], [(1,470,21,450),(2,469,22,449),(3,468,23,448),(4,467,24,447),(5,466,25,446),(6,465,26,445),(7,464,27,444),(8,463,28,443),(9,462,29,442),(10,461,30,441),(11,460,31,480),(12,459,32,479),(13,458,33,478),(14,457,34,477),(15,456,35,476),(16,455,36,475),(17,454,37,474),(18,453,38,473),(19,452,39,472),(20,451,40,471),(41,169,61,189),(42,168,62,188),(43,167,63,187),(44,166,64,186),(45,165,65,185),(46,164,66,184),(47,163,67,183),(48,162,68,182),(49,161,69,181),(50,200,70,180),(51,199,71,179),(52,198,72,178),(53,197,73,177),(54,196,74,176),(55,195,75,175),(56,194,76,174),(57,193,77,173),(58,192,78,172),(59,191,79,171),(60,190,80,170),(81,336,101,356),(82,335,102,355),(83,334,103,354),(84,333,104,353),(85,332,105,352),(86,331,106,351),(87,330,107,350),(88,329,108,349),(89,328,109,348),(90,327,110,347),(91,326,111,346),(92,325,112,345),(93,324,113,344),(94,323,114,343),(95,322,115,342),(96,321,116,341),(97,360,117,340),(98,359,118,339),(99,358,119,338),(100,357,120,337),(121,242,141,262),(122,241,142,261),(123,280,143,260),(124,279,144,259),(125,278,145,258),(126,277,146,257),(127,276,147,256),(128,275,148,255),(129,274,149,254),(130,273,150,253),(131,272,151,252),(132,271,152,251),(133,270,153,250),(134,269,154,249),(135,268,155,248),(136,267,156,247),(137,266,157,246),(138,265,158,245),(139,264,159,244),(140,263,160,243),(201,301,221,281),(202,300,222,320),(203,299,223,319),(204,298,224,318),(205,297,225,317),(206,296,226,316),(207,295,227,315),(208,294,228,314),(209,293,229,313),(210,292,230,312),(211,291,231,311),(212,290,232,310),(213,289,233,309),(214,288,234,308),(215,287,235,307),(216,286,236,306),(217,285,237,305),(218,284,238,304),(219,283,239,303),(220,282,240,302),(361,438,381,418),(362,437,382,417),(363,436,383,416),(364,435,384,415),(365,434,385,414),(366,433,386,413),(367,432,387,412),(368,431,388,411),(369,430,389,410),(370,429,390,409),(371,428,391,408),(372,427,392,407),(373,426,393,406),(374,425,394,405),(375,424,395,404),(376,423,396,403),(377,422,397,402),(378,421,398,401),(379,420,399,440),(380,419,400,439)]])
138 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
138 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | D5 | Q16 | D10 | D10 | C3×D4 | C3×D4 | C3×D5 | D20 | D20 | C3×Q16 | C6×D5 | C6×D5 | Dic20 | C3×D20 | C3×D20 | C3×Dic20 |
| kernel | C6×Dic20 | C3×Dic20 | C2×C120 | C6×Dic10 | C2×Dic20 | Dic20 | C2×C40 | C2×Dic10 | C60 | C2×C30 | C2×C24 | C30 | C24 | C2×C12 | C20 | C2×C10 | C2×C8 | C12 | C2×C6 | C10 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 2 | 4 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of C6×Dic20 ►in GL3(𝔽241) generated by
| 226 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 1 | 0 | 0 |
| 0 | 212 | 221 |
| 0 | 20 | 47 |
| 1 | 0 | 0 |
| 0 | 70 | 149 |
| 0 | 124 | 171 |
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