direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic62, C62⋊Q8, C4.11D62, C62.1C23, C22.8D62, C124.11C22, Dic31.1C22, C31⋊1(C2×Q8), (C2×C4).4D31, (C2×C124).4C2, (C2×C62).8C22, C2.3(C22×D31), (C2×Dic31).3C2, SmallGroup(496,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic62
G = < a,b,c | a2=b124=1, c2=b62, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic62
►Regular action on 496 points
Generators in S496
(1 163)(2 164)(3 165)(4 166)(5 167)(6 168)(7 169)(8 170)(9 171)(10 172)(11 173)(12 174)(13 175)(14 176)(15 177)(16 178)(17 179)(18 180)(19 181)(20 182)(21 183)(22 184)(23 185)(24 186)(25 187)(26 188)(27 189)(28 190)(29 191)(30 192)(31 193)(32 194)(33 195)(34 196)(35 197)(36 198)(37 199)(38 200)(39 201)(40 202)(41 203)(42 204)(43 205)(44 206)(45 207)(46 208)(47 209)(48 210)(49 211)(50 212)(51 213)(52 214)(53 215)(54 216)(55 217)(56 218)(57 219)(58 220)(59 221)(60 222)(61 223)(62 224)(63 225)(64 226)(65 227)(66 228)(67 229)(68 230)(69 231)(70 232)(71 233)(72 234)(73 235)(74 236)(75 237)(76 238)(77 239)(78 240)(79 241)(80 242)(81 243)(82 244)(83 245)(84 246)(85 247)(86 248)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 145)(108 146)(109 147)(110 148)(111 149)(112 150)(113 151)(114 152)(115 153)(116 154)(117 155)(118 156)(119 157)(120 158)(121 159)(122 160)(123 161)(124 162)(249 466)(250 467)(251 468)(252 469)(253 470)(254 471)(255 472)(256 473)(257 474)(258 475)(259 476)(260 477)(261 478)(262 479)(263 480)(264 481)(265 482)(266 483)(267 484)(268 485)(269 486)(270 487)(271 488)(272 489)(273 490)(274 491)(275 492)(276 493)(277 494)(278 495)(279 496)(280 373)(281 374)(282 375)(283 376)(284 377)(285 378)(286 379)(287 380)(288 381)(289 382)(290 383)(291 384)(292 385)(293 386)(294 387)(295 388)(296 389)(297 390)(298 391)(299 392)(300 393)(301 394)(302 395)(303 396)(304 397)(305 398)(306 399)(307 400)(308 401)(309 402)(310 403)(311 404)(312 405)(313 406)(314 407)(315 408)(316 409)(317 410)(318 411)(319 412)(320 413)(321 414)(322 415)(323 416)(324 417)(325 418)(326 419)(327 420)(328 421)(329 422)(330 423)(331 424)(332 425)(333 426)(334 427)(335 428)(336 429)(337 430)(338 431)(339 432)(340 433)(341 434)(342 435)(343 436)(344 437)(345 438)(346 439)(347 440)(348 441)(349 442)(350 443)(351 444)(352 445)(353 446)(354 447)(355 448)(356 449)(357 450)(358 451)(359 452)(360 453)(361 454)(362 455)(363 456)(364 457)(365 458)(366 459)(367 460)(368 461)(369 462)(370 463)(371 464)(372 465)
(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)(373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496)
(1 342 63 280)(2 341 64 279)(3 340 65 278)(4 339 66 277)(5 338 67 276)(6 337 68 275)(7 336 69 274)(8 335 70 273)(9 334 71 272)(10 333 72 271)(11 332 73 270)(12 331 74 269)(13 330 75 268)(14 329 76 267)(15 328 77 266)(16 327 78 265)(17 326 79 264)(18 325 80 263)(19 324 81 262)(20 323 82 261)(21 322 83 260)(22 321 84 259)(23 320 85 258)(24 319 86 257)(25 318 87 256)(26 317 88 255)(27 316 89 254)(28 315 90 253)(29 314 91 252)(30 313 92 251)(31 312 93 250)(32 311 94 249)(33 310 95 372)(34 309 96 371)(35 308 97 370)(36 307 98 369)(37 306 99 368)(38 305 100 367)(39 304 101 366)(40 303 102 365)(41 302 103 364)(42 301 104 363)(43 300 105 362)(44 299 106 361)(45 298 107 360)(46 297 108 359)(47 296 109 358)(48 295 110 357)(49 294 111 356)(50 293 112 355)(51 292 113 354)(52 291 114 353)(53 290 115 352)(54 289 116 351)(55 288 117 350)(56 287 118 349)(57 286 119 348)(58 285 120 347)(59 284 121 346)(60 283 122 345)(61 282 123 344)(62 281 124 343)(125 473 187 411)(126 472 188 410)(127 471 189 409)(128 470 190 408)(129 469 191 407)(130 468 192 406)(131 467 193 405)(132 466 194 404)(133 465 195 403)(134 464 196 402)(135 463 197 401)(136 462 198 400)(137 461 199 399)(138 460 200 398)(139 459 201 397)(140 458 202 396)(141 457 203 395)(142 456 204 394)(143 455 205 393)(144 454 206 392)(145 453 207 391)(146 452 208 390)(147 451 209 389)(148 450 210 388)(149 449 211 387)(150 448 212 386)(151 447 213 385)(152 446 214 384)(153 445 215 383)(154 444 216 382)(155 443 217 381)(156 442 218 380)(157 441 219 379)(158 440 220 378)(159 439 221 377)(160 438 222 376)(161 437 223 375)(162 436 224 374)(163 435 225 373)(164 434 226 496)(165 433 227 495)(166 432 228 494)(167 431 229 493)(168 430 230 492)(169 429 231 491)(170 428 232 490)(171 427 233 489)(172 426 234 488)(173 425 235 487)(174 424 236 486)(175 423 237 485)(176 422 238 484)(177 421 239 483)(178 420 240 482)(179 419 241 481)(180 418 242 480)(181 417 243 479)(182 416 244 478)(183 415 245 477)(184 414 246 476)(185 413 247 475)(186 412 248 474)
G:=sub<Sym(496)| (1,163)(2,164)(3,165)(4,166)(5,167)(6,168)(7,169)(8,170)(9,171)(10,172)(11,173)(12,174)(13,175)(14,176)(15,177)(16,178)(17,179)(18,180)(19,181)(20,182)(21,183)(22,184)(23,185)(24,186)(25,187)(26,188)(27,189)(28,190)(29,191)(30,192)(31,193)(32,194)(33,195)(34,196)(35,197)(36,198)(37,199)(38,200)(39,201)(40,202)(41,203)(42,204)(43,205)(44,206)(45,207)(46,208)(47,209)(48,210)(49,211)(50,212)(51,213)(52,214)(53,215)(54,216)(55,217)(56,218)(57,219)(58,220)(59,221)(60,222)(61,223)(62,224)(63,225)(64,226)(65,227)(66,228)(67,229)(68,230)(69,231)(70,232)(71,233)(72,234)(73,235)(74,236)(75,237)(76,238)(77,239)(78,240)(79,241)(80,242)(81,243)(82,244)(83,245)(84,246)(85,247)(86,248)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148)(111,149)(112,150)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158)(121,159)(122,160)(123,161)(124,162)(249,466)(250,467)(251,468)(252,469)(253,470)(254,471)(255,472)(256,473)(257,474)(258,475)(259,476)(260,477)(261,478)(262,479)(263,480)(264,481)(265,482)(266,483)(267,484)(268,485)(269,486)(270,487)(271,488)(272,489)(273,490)(274,491)(275,492)(276,493)(277,494)(278,495)(279,496)(280,373)(281,374)(282,375)(283,376)(284,377)(285,378)(286,379)(287,380)(288,381)(289,382)(290,383)(291,384)(292,385)(293,386)(294,387)(295,388)(296,389)(297,390)(298,391)(299,392)(300,393)(301,394)(302,395)(303,396)(304,397)(305,398)(306,399)(307,400)(308,401)(309,402)(310,403)(311,404)(312,405)(313,406)(314,407)(315,408)(316,409)(317,410)(318,411)(319,412)(320,413)(321,414)(322,415)(323,416)(324,417)(325,418)(326,419)(327,420)(328,421)(329,422)(330,423)(331,424)(332,425)(333,426)(334,427)(335,428)(336,429)(337,430)(338,431)(339,432)(340,433)(341,434)(342,435)(343,436)(344,437)(345,438)(346,439)(347,440)(348,441)(349,442)(350,443)(351,444)(352,445)(353,446)(354,447)(355,448)(356,449)(357,450)(358,451)(359,452)(360,453)(361,454)(362,455)(363,456)(364,457)(365,458)(366,459)(367,460)(368,461)(369,462)(370,463)(371,464)(372,465), (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)(373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,342,63,280)(2,341,64,279)(3,340,65,278)(4,339,66,277)(5,338,67,276)(6,337,68,275)(7,336,69,274)(8,335,70,273)(9,334,71,272)(10,333,72,271)(11,332,73,270)(12,331,74,269)(13,330,75,268)(14,329,76,267)(15,328,77,266)(16,327,78,265)(17,326,79,264)(18,325,80,263)(19,324,81,262)(20,323,82,261)(21,322,83,260)(22,321,84,259)(23,320,85,258)(24,319,86,257)(25,318,87,256)(26,317,88,255)(27,316,89,254)(28,315,90,253)(29,314,91,252)(30,313,92,251)(31,312,93,250)(32,311,94,249)(33,310,95,372)(34,309,96,371)(35,308,97,370)(36,307,98,369)(37,306,99,368)(38,305,100,367)(39,304,101,366)(40,303,102,365)(41,302,103,364)(42,301,104,363)(43,300,105,362)(44,299,106,361)(45,298,107,360)(46,297,108,359)(47,296,109,358)(48,295,110,357)(49,294,111,356)(50,293,112,355)(51,292,113,354)(52,291,114,353)(53,290,115,352)(54,289,116,351)(55,288,117,350)(56,287,118,349)(57,286,119,348)(58,285,120,347)(59,284,121,346)(60,283,122,345)(61,282,123,344)(62,281,124,343)(125,473,187,411)(126,472,188,410)(127,471,189,409)(128,470,190,408)(129,469,191,407)(130,468,192,406)(131,467,193,405)(132,466,194,404)(133,465,195,403)(134,464,196,402)(135,463,197,401)(136,462,198,400)(137,461,199,399)(138,460,200,398)(139,459,201,397)(140,458,202,396)(141,457,203,395)(142,456,204,394)(143,455,205,393)(144,454,206,392)(145,453,207,391)(146,452,208,390)(147,451,209,389)(148,450,210,388)(149,449,211,387)(150,448,212,386)(151,447,213,385)(152,446,214,384)(153,445,215,383)(154,444,216,382)(155,443,217,381)(156,442,218,380)(157,441,219,379)(158,440,220,378)(159,439,221,377)(160,438,222,376)(161,437,223,375)(162,436,224,374)(163,435,225,373)(164,434,226,496)(165,433,227,495)(166,432,228,494)(167,431,229,493)(168,430,230,492)(169,429,231,491)(170,428,232,490)(171,427,233,489)(172,426,234,488)(173,425,235,487)(174,424,236,486)(175,423,237,485)(176,422,238,484)(177,421,239,483)(178,420,240,482)(179,419,241,481)(180,418,242,480)(181,417,243,479)(182,416,244,478)(183,415,245,477)(184,414,246,476)(185,413,247,475)(186,412,248,474)>;
G:=Group( (1,163)(2,164)(3,165)(4,166)(5,167)(6,168)(7,169)(8,170)(9,171)(10,172)(11,173)(12,174)(13,175)(14,176)(15,177)(16,178)(17,179)(18,180)(19,181)(20,182)(21,183)(22,184)(23,185)(24,186)(25,187)(26,188)(27,189)(28,190)(29,191)(30,192)(31,193)(32,194)(33,195)(34,196)(35,197)(36,198)(37,199)(38,200)(39,201)(40,202)(41,203)(42,204)(43,205)(44,206)(45,207)(46,208)(47,209)(48,210)(49,211)(50,212)(51,213)(52,214)(53,215)(54,216)(55,217)(56,218)(57,219)(58,220)(59,221)(60,222)(61,223)(62,224)(63,225)(64,226)(65,227)(66,228)(67,229)(68,230)(69,231)(70,232)(71,233)(72,234)(73,235)(74,236)(75,237)(76,238)(77,239)(78,240)(79,241)(80,242)(81,243)(82,244)(83,245)(84,246)(85,247)(86,248)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,145)(108,146)(109,147)(110,148)(111,149)(112,150)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158)(121,159)(122,160)(123,161)(124,162)(249,466)(250,467)(251,468)(252,469)(253,470)(254,471)(255,472)(256,473)(257,474)(258,475)(259,476)(260,477)(261,478)(262,479)(263,480)(264,481)(265,482)(266,483)(267,484)(268,485)(269,486)(270,487)(271,488)(272,489)(273,490)(274,491)(275,492)(276,493)(277,494)(278,495)(279,496)(280,373)(281,374)(282,375)(283,376)(284,377)(285,378)(286,379)(287,380)(288,381)(289,382)(290,383)(291,384)(292,385)(293,386)(294,387)(295,388)(296,389)(297,390)(298,391)(299,392)(300,393)(301,394)(302,395)(303,396)(304,397)(305,398)(306,399)(307,400)(308,401)(309,402)(310,403)(311,404)(312,405)(313,406)(314,407)(315,408)(316,409)(317,410)(318,411)(319,412)(320,413)(321,414)(322,415)(323,416)(324,417)(325,418)(326,419)(327,420)(328,421)(329,422)(330,423)(331,424)(332,425)(333,426)(334,427)(335,428)(336,429)(337,430)(338,431)(339,432)(340,433)(341,434)(342,435)(343,436)(344,437)(345,438)(346,439)(347,440)(348,441)(349,442)(350,443)(351,444)(352,445)(353,446)(354,447)(355,448)(356,449)(357,450)(358,451)(359,452)(360,453)(361,454)(362,455)(363,456)(364,457)(365,458)(366,459)(367,460)(368,461)(369,462)(370,463)(371,464)(372,465), (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)(373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496), (1,342,63,280)(2,341,64,279)(3,340,65,278)(4,339,66,277)(5,338,67,276)(6,337,68,275)(7,336,69,274)(8,335,70,273)(9,334,71,272)(10,333,72,271)(11,332,73,270)(12,331,74,269)(13,330,75,268)(14,329,76,267)(15,328,77,266)(16,327,78,265)(17,326,79,264)(18,325,80,263)(19,324,81,262)(20,323,82,261)(21,322,83,260)(22,321,84,259)(23,320,85,258)(24,319,86,257)(25,318,87,256)(26,317,88,255)(27,316,89,254)(28,315,90,253)(29,314,91,252)(30,313,92,251)(31,312,93,250)(32,311,94,249)(33,310,95,372)(34,309,96,371)(35,308,97,370)(36,307,98,369)(37,306,99,368)(38,305,100,367)(39,304,101,366)(40,303,102,365)(41,302,103,364)(42,301,104,363)(43,300,105,362)(44,299,106,361)(45,298,107,360)(46,297,108,359)(47,296,109,358)(48,295,110,357)(49,294,111,356)(50,293,112,355)(51,292,113,354)(52,291,114,353)(53,290,115,352)(54,289,116,351)(55,288,117,350)(56,287,118,349)(57,286,119,348)(58,285,120,347)(59,284,121,346)(60,283,122,345)(61,282,123,344)(62,281,124,343)(125,473,187,411)(126,472,188,410)(127,471,189,409)(128,470,190,408)(129,469,191,407)(130,468,192,406)(131,467,193,405)(132,466,194,404)(133,465,195,403)(134,464,196,402)(135,463,197,401)(136,462,198,400)(137,461,199,399)(138,460,200,398)(139,459,201,397)(140,458,202,396)(141,457,203,395)(142,456,204,394)(143,455,205,393)(144,454,206,392)(145,453,207,391)(146,452,208,390)(147,451,209,389)(148,450,210,388)(149,449,211,387)(150,448,212,386)(151,447,213,385)(152,446,214,384)(153,445,215,383)(154,444,216,382)(155,443,217,381)(156,442,218,380)(157,441,219,379)(158,440,220,378)(159,439,221,377)(160,438,222,376)(161,437,223,375)(162,436,224,374)(163,435,225,373)(164,434,226,496)(165,433,227,495)(166,432,228,494)(167,431,229,493)(168,430,230,492)(169,429,231,491)(170,428,232,490)(171,427,233,489)(172,426,234,488)(173,425,235,487)(174,424,236,486)(175,423,237,485)(176,422,238,484)(177,421,239,483)(178,420,240,482)(179,419,241,481)(180,418,242,480)(181,417,243,479)(182,416,244,478)(183,415,245,477)(184,414,246,476)(185,413,247,475)(186,412,248,474) );
G=PermutationGroup([[(1,163),(2,164),(3,165),(4,166),(5,167),(6,168),(7,169),(8,170),(9,171),(10,172),(11,173),(12,174),(13,175),(14,176),(15,177),(16,178),(17,179),(18,180),(19,181),(20,182),(21,183),(22,184),(23,185),(24,186),(25,187),(26,188),(27,189),(28,190),(29,191),(30,192),(31,193),(32,194),(33,195),(34,196),(35,197),(36,198),(37,199),(38,200),(39,201),(40,202),(41,203),(42,204),(43,205),(44,206),(45,207),(46,208),(47,209),(48,210),(49,211),(50,212),(51,213),(52,214),(53,215),(54,216),(55,217),(56,218),(57,219),(58,220),(59,221),(60,222),(61,223),(62,224),(63,225),(64,226),(65,227),(66,228),(67,229),(68,230),(69,231),(70,232),(71,233),(72,234),(73,235),(74,236),(75,237),(76,238),(77,239),(78,240),(79,241),(80,242),(81,243),(82,244),(83,245),(84,246),(85,247),(86,248),(87,125),(88,126),(89,127),(90,128),(91,129),(92,130),(93,131),(94,132),(95,133),(96,134),(97,135),(98,136),(99,137),(100,138),(101,139),(102,140),(103,141),(104,142),(105,143),(106,144),(107,145),(108,146),(109,147),(110,148),(111,149),(112,150),(113,151),(114,152),(115,153),(116,154),(117,155),(118,156),(119,157),(120,158),(121,159),(122,160),(123,161),(124,162),(249,466),(250,467),(251,468),(252,469),(253,470),(254,471),(255,472),(256,473),(257,474),(258,475),(259,476),(260,477),(261,478),(262,479),(263,480),(264,481),(265,482),(266,483),(267,484),(268,485),(269,486),(270,487),(271,488),(272,489),(273,490),(274,491),(275,492),(276,493),(277,494),(278,495),(279,496),(280,373),(281,374),(282,375),(283,376),(284,377),(285,378),(286,379),(287,380),(288,381),(289,382),(290,383),(291,384),(292,385),(293,386),(294,387),(295,388),(296,389),(297,390),(298,391),(299,392),(300,393),(301,394),(302,395),(303,396),(304,397),(305,398),(306,399),(307,400),(308,401),(309,402),(310,403),(311,404),(312,405),(313,406),(314,407),(315,408),(316,409),(317,410),(318,411),(319,412),(320,413),(321,414),(322,415),(323,416),(324,417),(325,418),(326,419),(327,420),(328,421),(329,422),(330,423),(331,424),(332,425),(333,426),(334,427),(335,428),(336,429),(337,430),(338,431),(339,432),(340,433),(341,434),(342,435),(343,436),(344,437),(345,438),(346,439),(347,440),(348,441),(349,442),(350,443),(351,444),(352,445),(353,446),(354,447),(355,448),(356,449),(357,450),(358,451),(359,452),(360,453),(361,454),(362,455),(363,456),(364,457),(365,458),(366,459),(367,460),(368,461),(369,462),(370,463),(371,464),(372,465)], [(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),(373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496)], [(1,342,63,280),(2,341,64,279),(3,340,65,278),(4,339,66,277),(5,338,67,276),(6,337,68,275),(7,336,69,274),(8,335,70,273),(9,334,71,272),(10,333,72,271),(11,332,73,270),(12,331,74,269),(13,330,75,268),(14,329,76,267),(15,328,77,266),(16,327,78,265),(17,326,79,264),(18,325,80,263),(19,324,81,262),(20,323,82,261),(21,322,83,260),(22,321,84,259),(23,320,85,258),(24,319,86,257),(25,318,87,256),(26,317,88,255),(27,316,89,254),(28,315,90,253),(29,314,91,252),(30,313,92,251),(31,312,93,250),(32,311,94,249),(33,310,95,372),(34,309,96,371),(35,308,97,370),(36,307,98,369),(37,306,99,368),(38,305,100,367),(39,304,101,366),(40,303,102,365),(41,302,103,364),(42,301,104,363),(43,300,105,362),(44,299,106,361),(45,298,107,360),(46,297,108,359),(47,296,109,358),(48,295,110,357),(49,294,111,356),(50,293,112,355),(51,292,113,354),(52,291,114,353),(53,290,115,352),(54,289,116,351),(55,288,117,350),(56,287,118,349),(57,286,119,348),(58,285,120,347),(59,284,121,346),(60,283,122,345),(61,282,123,344),(62,281,124,343),(125,473,187,411),(126,472,188,410),(127,471,189,409),(128,470,190,408),(129,469,191,407),(130,468,192,406),(131,467,193,405),(132,466,194,404),(133,465,195,403),(134,464,196,402),(135,463,197,401),(136,462,198,400),(137,461,199,399),(138,460,200,398),(139,459,201,397),(140,458,202,396),(141,457,203,395),(142,456,204,394),(143,455,205,393),(144,454,206,392),(145,453,207,391),(146,452,208,390),(147,451,209,389),(148,450,210,388),(149,449,211,387),(150,448,212,386),(151,447,213,385),(152,446,214,384),(153,445,215,383),(154,444,216,382),(155,443,217,381),(156,442,218,380),(157,441,219,379),(158,440,220,378),(159,439,221,377),(160,438,222,376),(161,437,223,375),(162,436,224,374),(163,435,225,373),(164,434,226,496),(165,433,227,495),(166,432,228,494),(167,431,229,493),(168,430,230,492),(169,429,231,491),(170,428,232,490),(171,427,233,489),(172,426,234,488),(173,425,235,487),(174,424,236,486),(175,423,237,485),(176,422,238,484),(177,421,239,483),(178,420,240,482),(179,419,241,481),(180,418,242,480),(181,417,243,479),(182,416,244,478),(183,415,245,477),(184,414,246,476),(185,413,247,475),(186,412,248,474)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 31A | ··· | 31O | 62A | ··· | 62AS | 124A | ··· | 124BH |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 62 | 62 | 62 | 62 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D31 | D62 | D62 | Dic62 |
| kernel | C2×Dic62 | Dic62 | C2×Dic31 | C2×C124 | C62 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 15 | 30 | 15 | 60 |
Matrix representation of C2×Dic62 ►in GL3(𝔽373) generated by
| 372 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 372 | 0 | 0 |
| 0 | 237 | 84 |
| 0 | 310 | 283 |
| 372 | 0 | 0 |
| 0 | 87 | 125 |
| 0 | 208 | 286 |
G:=sub<GL(3,GF(373))| [372,0,0,0,1,0,0,0,1],[372,0,0,0,237,310,0,84,283],[372,0,0,0,87,208,0,125,286] >;
C2×Dic62 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{62} % in TeX
G:=Group("C2xDic62"); // GroupNames label
G:=SmallGroup(496,27);
// by ID
G=gap.SmallGroup(496,27);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,40,182,42,12004]);
// Polycyclic
G:=Group<a,b,c|a^2=b^124=1,c^2=b^62,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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