Copied to
clipboard

G = C14.(C4×D4)  order 448 = 26·7

4th non-split extension by C14 of C4×D4 acting via C4×D4/C22⋊C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C14.(C4×D4)
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×C4×Dic7 — C14.(C4×D4)
 Lower central C7 — C2×C14 — C14.(C4×D4)
 Upper central C1 — C23 — C2.C42

Generators and relations for C14.(C4×D4)
G = < a,b,c,d | a14=b4=c4=1, d2=a7, bab-1=cac-1=a-1, ad=da, cbc-1=dbd-1=a7b, dcd-1=c-1 >

Subgroups: 604 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.63C23, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C22×Dic7, C22×C28, C14.C42, C7×C2.C42, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C14.(C4×D4)
Quotients:

Smallest permutation representation of C14.(C4×D4)
Regular action on 448 points
Generators in S448
(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)
(1 315 152 249)(2 314 153 248)(3 313 154 247)(4 312 141 246)(5 311 142 245)(6 310 143 244)(7 309 144 243)(8 322 145 242)(9 321 146 241)(10 320 147 240)(11 319 148 239)(12 318 149 252)(13 317 150 251)(14 316 151 250)(15 398 257 136)(16 397 258 135)(17 396 259 134)(18 395 260 133)(19 394 261 132)(20 393 262 131)(21 406 263 130)(22 405 264 129)(23 404 265 128)(24 403 266 127)(25 402 253 140)(26 401 254 139)(27 400 255 138)(28 399 256 137)(29 307 156 211)(30 306 157 224)(31 305 158 223)(32 304 159 222)(33 303 160 221)(34 302 161 220)(35 301 162 219)(36 300 163 218)(37 299 164 217)(38 298 165 216)(39 297 166 215)(40 296 167 214)(41 295 168 213)(42 308 155 212)(43 334 358 210)(44 333 359 209)(45 332 360 208)(46 331 361 207)(47 330 362 206)(48 329 363 205)(49 328 364 204)(50 327 351 203)(51 326 352 202)(52 325 353 201)(53 324 354 200)(54 323 355 199)(55 336 356 198)(56 335 357 197)(57 82 231 417)(58 81 232 416)(59 80 233 415)(60 79 234 414)(61 78 235 413)(62 77 236 412)(63 76 237 411)(64 75 238 410)(65 74 225 409)(66 73 226 408)(67 72 227 407)(68 71 228 420)(69 84 229 419)(70 83 230 418)(85 271 427 175)(86 270 428 174)(87 269 429 173)(88 268 430 172)(89 267 431 171)(90 280 432 170)(91 279 433 169)(92 278 434 182)(93 277 421 181)(94 276 422 180)(95 275 423 179)(96 274 424 178)(97 273 425 177)(98 272 426 176)(99 119 371 338)(100 118 372 337)(101 117 373 350)(102 116 374 349)(103 115 375 348)(104 114 376 347)(105 113 377 346)(106 126 378 345)(107 125 365 344)(108 124 366 343)(109 123 367 342)(110 122 368 341)(111 121 369 340)(112 120 370 339)(183 287 440 392)(184 286 441 391)(185 285 442 390)(186 284 443 389)(187 283 444 388)(188 282 445 387)(189 281 446 386)(190 294 447 385)(191 293 448 384)(192 292 435 383)(193 291 436 382)(194 290 437 381)(195 289 438 380)(196 288 439 379)
(1 448 95 375)(2 447 96 374)(3 446 97 373)(4 445 98 372)(5 444 85 371)(6 443 86 370)(7 442 87 369)(8 441 88 368)(9 440 89 367)(10 439 90 366)(11 438 91 365)(12 437 92 378)(13 436 93 377)(14 435 94 376)(15 73 47 40)(16 72 48 39)(17 71 49 38)(18 84 50 37)(19 83 51 36)(20 82 52 35)(21 81 53 34)(22 80 54 33)(23 79 55 32)(24 78 56 31)(25 77 43 30)(26 76 44 29)(27 75 45 42)(28 74 46 41)(57 208 219 138)(58 207 220 137)(59 206 221 136)(60 205 222 135)(61 204 223 134)(62 203 224 133)(63 202 211 132)(64 201 212 131)(65 200 213 130)(66 199 214 129)(67 198 215 128)(68 197 216 127)(69 210 217 140)(70 209 218 139)(99 142 187 427)(100 141 188 426)(101 154 189 425)(102 153 190 424)(103 152 191 423)(104 151 192 422)(105 150 193 421)(106 149 194 434)(107 148 195 433)(108 147 196 432)(109 146 183 431)(110 145 184 430)(111 144 185 429)(112 143 186 428)(113 244 291 174)(114 243 292 173)(115 242 293 172)(116 241 294 171)(117 240 281 170)(118 239 282 169)(119 252 283 182)(120 251 284 181)(121 250 285 180)(122 249 286 179)(123 248 287 178)(124 247 288 177)(125 246 289 176)(126 245 290 175)(155 255 410 360)(156 254 411 359)(157 253 412 358)(158 266 413 357)(159 265 414 356)(160 264 415 355)(161 263 416 354)(162 262 417 353)(163 261 418 352)(164 260 419 351)(165 259 420 364)(166 258 407 363)(167 257 408 362)(168 256 409 361)(225 324 295 406)(226 323 296 405)(227 336 297 404)(228 335 298 403)(229 334 299 402)(230 333 300 401)(231 332 301 400)(232 331 302 399)(233 330 303 398)(234 329 304 397)(235 328 305 396)(236 327 306 395)(237 326 307 394)(238 325 308 393)(267 349 321 385)(268 348 322 384)(269 347 309 383)(270 346 310 382)(271 345 311 381)(272 344 312 380)(273 343 313 379)(274 342 314 392)(275 341 315 391)(276 340 316 390)(277 339 317 389)(278 338 318 388)(279 337 319 387)(280 350 320 386)
(1 223 8 216)(2 224 9 217)(3 211 10 218)(4 212 11 219)(5 213 12 220)(6 214 13 221)(7 215 14 222)(15 346 22 339)(16 347 23 340)(17 348 24 341)(18 349 25 342)(19 350 26 343)(20 337 27 344)(21 338 28 345)(29 313 36 320)(30 314 37 321)(31 315 38 322)(32 316 39 309)(33 317 40 310)(34 318 41 311)(35 319 42 312)(43 392 50 385)(44 379 51 386)(45 380 52 387)(46 381 53 388)(47 382 54 389)(48 383 55 390)(49 384 56 391)(57 98 64 91)(58 85 65 92)(59 86 66 93)(60 87 67 94)(61 88 68 95)(62 89 69 96)(63 90 70 97)(71 268 78 275)(72 269 79 276)(73 270 80 277)(74 271 81 278)(75 272 82 279)(76 273 83 280)(77 274 84 267)(99 406 106 399)(100 393 107 400)(101 394 108 401)(102 395 109 402)(103 396 110 403)(104 397 111 404)(105 398 112 405)(113 264 120 257)(114 265 121 258)(115 266 122 259)(116 253 123 260)(117 254 124 261)(118 255 125 262)(119 256 126 263)(127 375 134 368)(128 376 135 369)(129 377 136 370)(130 378 137 371)(131 365 138 372)(132 366 139 373)(133 367 140 374)(141 308 148 301)(142 295 149 302)(143 296 150 303)(144 297 151 304)(145 298 152 305)(146 299 153 306)(147 300 154 307)(155 246 162 239)(156 247 163 240)(157 248 164 241)(158 249 165 242)(159 250 166 243)(160 251 167 244)(161 252 168 245)(169 410 176 417)(170 411 177 418)(171 412 178 419)(172 413 179 420)(173 414 180 407)(174 415 181 408)(175 416 182 409)(183 334 190 327)(184 335 191 328)(185 336 192 329)(186 323 193 330)(187 324 194 331)(188 325 195 332)(189 326 196 333)(197 448 204 441)(198 435 205 442)(199 436 206 443)(200 437 207 444)(201 438 208 445)(202 439 209 446)(203 440 210 447)(225 434 232 427)(226 421 233 428)(227 422 234 429)(228 423 235 430)(229 424 236 431)(230 425 237 432)(231 426 238 433)(281 359 288 352)(282 360 289 353)(283 361 290 354)(284 362 291 355)(285 363 292 356)(286 364 293 357)(287 351 294 358)

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

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

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

88 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 7A 7B 7C 14A ··· 14U 28A ··· 28AJ order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 2 2 2 4 4 4 4 14 ··· 14 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + + - + - + image C1 C2 C2 C2 C2 C2 C4 D4 Q8 D7 C4○D4 D14 Dic14 C4×D7 C4○D28 D4×D7 D4⋊2D7 Q8⋊2D7 kernel C14.(C4×D4) C14.C42 C7×C2.C42 C2×C4×Dic7 C2×Dic7⋊C4 C2×C4⋊Dic7 Dic7⋊C4 C2×Dic7 C2×C28 C2.C42 C2×C14 C22×C4 C2×C4 C2×C4 C22 C22 C22 C22 # reps 1 3 1 1 1 1 8 2 2 3 8 9 12 12 12 3 6 3

Matrix representation of C14.(C4×D4) in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 21 21 0 0 0 0 8 26
,
 21 14 0 0 0 0 14 8 0 0 0 0 0 0 18 5 0 0 0 0 5 11 0 0 0 0 0 0 27 1 0 0 0 0 24 2
,
 14 8 0 0 0 0 8 15 0 0 0 0 0 0 5 11 0 0 0 0 11 24 0 0 0 0 0 0 24 17 0 0 0 0 2 5
,
 0 28 0 0 0 0 1 0 0 0 0 0 0 0 27 13 0 0 0 0 13 2 0 0 0 0 0 0 28 0 0 0 0 0 0 28

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,21,8,0,0,0,0,21,26],[21,14,0,0,0,0,14,8,0,0,0,0,0,0,18,5,0,0,0,0,5,11,0,0,0,0,0,0,27,24,0,0,0,0,1,2],[14,8,0,0,0,0,8,15,0,0,0,0,0,0,5,11,0,0,0,0,11,24,0,0,0,0,0,0,24,2,0,0,0,0,17,5],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,27,13,0,0,0,0,13,2,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;

C14.(C4×D4) in GAP, Magma, Sage, TeX

C_{14}.(C_4\times D_4)
% in TeX

G:=Group("C14.(C4xD4)");
// GroupNames label

G:=SmallGroup(448,189);
// by ID

G=gap.SmallGroup(448,189);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,64,926,219,268,18822]);
// Polycyclic

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

׿
×
𝔽