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## G = C2×C14.C42order 448 = 26·7

### Direct product of C2 and C14.C42

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×C14.C42
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C23×Dic7 — C2×C14.C42
 Lower central C7 — C14 — C2×C14.C42
 Upper central C1 — C24 — C23×C4

Generators and relations for C2×C14.C42
G = < a,b,c,d | a2=b14=c4=d4=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b7c >

Subgroups: 1092 in 330 conjugacy classes, 183 normal (21 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C23, C14, C14, C22×C4, C22×C4, C24, Dic7, C28, C2×C14, C2×C14, C2.C42, C23×C4, C23×C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C22×C14, C2×C2.C42, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C23×C14, C14.C42, C23×Dic7, C23×C28, C2×C14.C42
Quotients:

Smallest permutation representation of C2×C14.C42
Regular action on 448 points
Generators in S448
(1 128)(2 129)(3 130)(4 131)(5 132)(6 133)(7 134)(8 135)(9 136)(10 137)(11 138)(12 139)(13 140)(14 127)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 163)(30 164)(31 165)(32 166)(33 167)(34 168)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(41 161)(42 162)(43 148)(44 149)(45 150)(46 151)(47 152)(48 153)(49 154)(50 141)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 189)(58 190)(59 191)(60 192)(61 193)(62 194)(63 195)(64 196)(65 183)(66 184)(67 185)(68 186)(69 187)(70 188)(71 179)(72 180)(73 181)(74 182)(75 169)(76 170)(77 171)(78 172)(79 173)(80 174)(81 175)(82 176)(83 177)(84 178)(85 217)(86 218)(87 219)(88 220)(89 221)(90 222)(91 223)(92 224)(93 211)(94 212)(95 213)(96 214)(97 215)(98 216)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)(105 209)(106 210)(107 197)(108 198)(109 199)(110 200)(111 201)(112 202)(225 362)(226 363)(227 364)(228 351)(229 352)(230 353)(231 354)(232 355)(233 356)(234 357)(235 358)(236 359)(237 360)(238 361)(239 343)(240 344)(241 345)(242 346)(243 347)(244 348)(245 349)(246 350)(247 337)(248 338)(249 339)(250 340)(251 341)(252 342)(253 391)(254 392)(255 379)(256 380)(257 381)(258 382)(259 383)(260 384)(261 385)(262 386)(263 387)(264 388)(265 389)(266 390)(267 367)(268 368)(269 369)(270 370)(271 371)(272 372)(273 373)(274 374)(275 375)(276 376)(277 377)(278 378)(279 365)(280 366)(281 417)(282 418)(283 419)(284 420)(285 407)(286 408)(287 409)(288 410)(289 411)(290 412)(291 413)(292 414)(293 415)(294 416)(295 394)(296 395)(297 396)(298 397)(299 398)(300 399)(301 400)(302 401)(303 402)(304 403)(305 404)(306 405)(307 406)(308 393)(309 447)(310 448)(311 435)(312 436)(313 437)(314 438)(315 439)(316 440)(317 441)(318 442)(319 443)(320 444)(321 445)(322 446)(323 428)(324 429)(325 430)(326 431)(327 432)(328 433)(329 434)(330 421)(331 422)(332 423)(333 424)(334 425)(335 426)(336 427)
(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)
(1 265 40 225)(2 264 41 238)(3 263 42 237)(4 262 29 236)(5 261 30 235)(6 260 31 234)(7 259 32 233)(8 258 33 232)(9 257 34 231)(10 256 35 230)(11 255 36 229)(12 254 37 228)(13 253 38 227)(14 266 39 226)(15 268 44 249)(16 267 45 248)(17 280 46 247)(18 279 47 246)(19 278 48 245)(20 277 49 244)(21 276 50 243)(22 275 51 242)(23 274 52 241)(24 273 53 240)(25 272 54 239)(26 271 55 252)(27 270 56 251)(28 269 43 250)(57 310 97 293)(58 309 98 292)(59 322 85 291)(60 321 86 290)(61 320 87 289)(62 319 88 288)(63 318 89 287)(64 317 90 286)(65 316 91 285)(66 315 92 284)(67 314 93 283)(68 313 94 282)(69 312 95 281)(70 311 96 294)(71 332 104 295)(72 331 105 308)(73 330 106 307)(74 329 107 306)(75 328 108 305)(76 327 109 304)(77 326 110 303)(78 325 111 302)(79 324 112 301)(80 323 99 300)(81 336 100 299)(82 335 101 298)(83 334 102 297)(84 333 103 296)(113 376 141 347)(114 375 142 346)(115 374 143 345)(116 373 144 344)(117 372 145 343)(118 371 146 342)(119 370 147 341)(120 369 148 340)(121 368 149 339)(122 367 150 338)(123 366 151 337)(124 365 152 350)(125 378 153 349)(126 377 154 348)(127 390 159 363)(128 389 160 362)(129 388 161 361)(130 387 162 360)(131 386 163 359)(132 385 164 358)(133 384 165 357)(134 383 166 356)(135 382 167 355)(136 381 168 354)(137 380 155 353)(138 379 156 352)(139 392 157 351)(140 391 158 364)(169 433 198 404)(170 432 199 403)(171 431 200 402)(172 430 201 401)(173 429 202 400)(174 428 203 399)(175 427 204 398)(176 426 205 397)(177 425 206 396)(178 424 207 395)(179 423 208 394)(180 422 209 393)(181 421 210 406)(182 434 197 405)(183 440 223 407)(184 439 224 420)(185 438 211 419)(186 437 212 418)(187 436 213 417)(188 435 214 416)(189 448 215 415)(190 447 216 414)(191 446 217 413)(192 445 218 412)(193 444 219 411)(194 443 220 410)(195 442 221 409)(196 441 222 408)
(1 193 24 180)(2 194 25 181)(3 195 26 182)(4 196 27 169)(5 183 28 170)(6 184 15 171)(7 185 16 172)(8 186 17 173)(9 187 18 174)(10 188 19 175)(11 189 20 176)(12 190 21 177)(13 191 22 178)(14 192 23 179)(29 222 56 198)(30 223 43 199)(31 224 44 200)(32 211 45 201)(33 212 46 202)(34 213 47 203)(35 214 48 204)(36 215 49 205)(37 216 50 206)(38 217 51 207)(39 218 52 208)(40 219 53 209)(41 220 54 210)(42 221 55 197)(57 126 82 138)(58 113 83 139)(59 114 84 140)(60 115 71 127)(61 116 72 128)(62 117 73 129)(63 118 74 130)(64 119 75 131)(65 120 76 132)(66 121 77 133)(67 122 78 134)(68 123 79 135)(69 124 80 136)(70 125 81 137)(85 142 103 158)(86 143 104 159)(87 144 105 160)(88 145 106 161)(89 146 107 162)(90 147 108 163)(91 148 109 164)(92 149 110 165)(93 150 111 166)(94 151 112 167)(95 152 99 168)(96 153 100 155)(97 154 101 156)(98 141 102 157)(225 418 240 400)(226 419 241 401)(227 420 242 402)(228 407 243 403)(229 408 244 404)(230 409 245 405)(231 410 246 406)(232 411 247 393)(233 412 248 394)(234 413 249 395)(235 414 250 396)(236 415 251 397)(237 416 252 398)(238 417 239 399)(253 439 275 431)(254 440 276 432)(255 441 277 433)(256 442 278 434)(257 443 279 421)(258 444 280 422)(259 445 267 423)(260 446 268 424)(261 447 269 425)(262 448 270 426)(263 435 271 427)(264 436 272 428)(265 437 273 429)(266 438 274 430)(281 343 300 361)(282 344 301 362)(283 345 302 363)(284 346 303 364)(285 347 304 351)(286 348 305 352)(287 349 306 353)(288 350 307 354)(289 337 308 355)(290 338 295 356)(291 339 296 357)(292 340 297 358)(293 341 298 359)(294 342 299 360)(309 369 334 385)(310 370 335 386)(311 371 336 387)(312 372 323 388)(313 373 324 389)(314 374 325 390)(315 375 326 391)(316 376 327 392)(317 377 328 379)(318 378 329 380)(319 365 330 381)(320 366 331 382)(321 367 332 383)(322 368 333 384)

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

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

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

136 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4H 4I ··· 4X 7A 7B 7C 14A ··· 14AS 28A ··· 28AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 ··· 2 14 ··· 14 2 2 2 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - + + - + image C1 C2 C2 C2 C4 C4 D4 Q8 D7 Dic7 D14 D14 Dic14 C4×D7 D28 C7⋊D4 kernel C2×C14.C42 C14.C42 C23×Dic7 C23×C28 C22×Dic7 C22×C28 C22×C14 C22×C14 C23×C4 C22×C4 C22×C4 C24 C23 C23 C23 C23 # reps 1 4 2 1 16 8 6 2 3 12 6 3 12 24 12 24

Matrix representation of C2×C14.C42 in GL5(𝔽29)

 1 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 3 21 0 0 0 8 8
,
 17 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 17 7 0 0 0 0 12
,
 28 0 0 0 0 0 12 0 0 0 0 0 28 0 0 0 0 0 5 16 0 0 0 13 24

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,8,0,0,0,21,8],[17,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,17,0,0,0,0,7,12],[28,0,0,0,0,0,12,0,0,0,0,0,28,0,0,0,0,0,5,13,0,0,0,16,24] >;

C2×C14.C42 in GAP, Magma, Sage, TeX

C_2\times C_{14}.C_4^2
% in TeX

G:=Group("C2xC14.C4^2");
// GroupNames label

G:=SmallGroup(448,742);
// by ID

G=gap.SmallGroup(448,742);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,758,100,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^14=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^7*c>;
// generators/relations

׿
×
𝔽