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## G = C14×C8⋊C4order 448 = 26·7

### Direct product of C14 and C8⋊C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C14×C8⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C28 — C2×C56 — C7×C8⋊C4 — C14×C8⋊C4
 Lower central C1 — C2 — C14×C8⋊C4
 Upper central C1 — C22×C28 — C14×C8⋊C4

Generators and relations for C14×C8⋊C4
G = < a,b,c | a14=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 162 in 146 conjugacy classes, 130 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C22×C4, C22×C4, C28, C28, C2×C14, C2×C14, C8⋊C4, C2×C42, C22×C8, C56, C2×C28, C2×C28, C2×C28, C22×C14, C2×C8⋊C4, C4×C28, C2×C56, C22×C28, C22×C28, C7×C8⋊C4, C2×C4×C28, C22×C56, C14×C8⋊C4
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C42, M4(2), C22×C4, C28, C2×C14, C8⋊C4, C2×C42, C2×M4(2), C2×C28, C22×C14, C2×C8⋊C4, C4×C28, C7×M4(2), C22×C28, C7×C8⋊C4, C2×C4×C28, C14×M4(2), C14×C8⋊C4

Smallest permutation representation of C14×C8⋊C4
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)
(1 386 113 207 247 175 320 410)(2 387 114 208 248 176 321 411)(3 388 115 209 249 177 322 412)(4 389 116 210 250 178 309 413)(5 390 117 197 251 179 310 414)(6 391 118 198 252 180 311 415)(7 392 119 199 239 181 312 416)(8 379 120 200 240 182 313 417)(9 380 121 201 241 169 314 418)(10 381 122 202 242 170 315 419)(11 382 123 203 243 171 316 420)(12 383 124 204 244 172 317 407)(13 384 125 205 245 173 318 408)(14 385 126 206 246 174 319 409)(15 376 363 273 41 53 421 348)(16 377 364 274 42 54 422 349)(17 378 351 275 29 55 423 350)(18 365 352 276 30 56 424 337)(19 366 353 277 31 43 425 338)(20 367 354 278 32 44 426 339)(21 368 355 279 33 45 427 340)(22 369 356 280 34 46 428 341)(23 370 357 267 35 47 429 342)(24 371 358 268 36 48 430 343)(25 372 359 269 37 49 431 344)(26 373 360 270 38 50 432 345)(27 374 361 271 39 51 433 346)(28 375 362 272 40 52 434 347)(57 437 293 151 333 87 107 189)(58 438 294 152 334 88 108 190)(59 439 281 153 335 89 109 191)(60 440 282 154 336 90 110 192)(61 441 283 141 323 91 111 193)(62 442 284 142 324 92 112 194)(63 443 285 143 325 93 99 195)(64 444 286 144 326 94 100 196)(65 445 287 145 327 95 101 183)(66 446 288 146 328 96 102 184)(67 447 289 147 329 97 103 185)(68 448 290 148 330 98 104 186)(69 435 291 149 331 85 105 187)(70 436 292 150 332 86 106 188)(71 219 167 261 236 398 138 307)(72 220 168 262 237 399 139 308)(73 221 155 263 238 400 140 295)(74 222 156 264 225 401 127 296)(75 223 157 265 226 402 128 297)(76 224 158 266 227 403 129 298)(77 211 159 253 228 404 130 299)(78 212 160 254 229 405 131 300)(79 213 161 255 230 406 132 301)(80 214 162 256 231 393 133 302)(81 215 163 257 232 394 134 303)(82 216 164 258 233 395 135 304)(83 217 165 259 234 396 136 305)(84 218 166 260 235 397 137 306)
(1 438 352 135)(2 439 353 136)(3 440 354 137)(4 441 355 138)(5 442 356 139)(6 443 357 140)(7 444 358 127)(8 445 359 128)(9 446 360 129)(10 447 361 130)(11 448 362 131)(12 435 363 132)(13 436 364 133)(14 437 351 134)(15 230 317 187)(16 231 318 188)(17 232 319 189)(18 233 320 190)(19 234 321 191)(20 235 322 192)(21 236 309 193)(22 237 310 194)(23 238 311 195)(24 225 312 196)(25 226 313 183)(26 227 314 184)(27 228 315 185)(28 229 316 186)(29 81 126 151)(30 82 113 152)(31 83 114 153)(32 84 115 154)(33 71 116 141)(34 72 117 142)(35 73 118 143)(36 74 119 144)(37 75 120 145)(38 76 121 146)(39 77 122 147)(40 78 123 148)(41 79 124 149)(42 80 125 150)(43 396 208 59)(44 397 209 60)(45 398 210 61)(46 399 197 62)(47 400 198 63)(48 401 199 64)(49 402 200 65)(50 403 201 66)(51 404 202 67)(52 405 203 68)(53 406 204 69)(54 393 205 70)(55 394 206 57)(56 395 207 58)(85 421 161 244)(86 422 162 245)(87 423 163 246)(88 424 164 247)(89 425 165 248)(90 426 166 249)(91 427 167 250)(92 428 168 251)(93 429 155 252)(94 430 156 239)(95 431 157 240)(96 432 158 241)(97 433 159 242)(98 434 160 243)(99 267 263 391)(100 268 264 392)(101 269 265 379)(102 270 266 380)(103 271 253 381)(104 272 254 382)(105 273 255 383)(106 274 256 384)(107 275 257 385)(108 276 258 386)(109 277 259 387)(110 278 260 388)(111 279 261 389)(112 280 262 390)(169 288 345 298)(170 289 346 299)(171 290 347 300)(172 291 348 301)(173 292 349 302)(174 293 350 303)(175 294 337 304)(176 281 338 305)(177 282 339 306)(178 283 340 307)(179 284 341 308)(180 285 342 295)(181 286 343 296)(182 287 344 297)(211 419 329 374)(212 420 330 375)(213 407 331 376)(214 408 332 377)(215 409 333 378)(216 410 334 365)(217 411 335 366)(218 412 336 367)(219 413 323 368)(220 414 324 369)(221 415 325 370)(222 416 326 371)(223 417 327 372)(224 418 328 373)

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

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

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

280 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 7A ··· 7F 8A ··· 8P 14A ··· 14AP 28A ··· 28AV 28AW ··· 28CR 56A ··· 56CR order 1 2 ··· 2 4 ··· 4 4 ··· 4 7 ··· 7 8 ··· 8 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

280 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C4 C7 C14 C14 C14 C28 C28 C28 M4(2) C7×M4(2) kernel C14×C8⋊C4 C7×C8⋊C4 C2×C4×C28 C22×C56 C4×C28 C2×C56 C22×C28 C2×C8⋊C4 C8⋊C4 C2×C42 C22×C8 C42 C2×C8 C22×C4 C2×C14 C22 # reps 1 4 1 2 4 16 4 6 24 6 12 24 96 24 8 48

Matrix representation of C14×C8⋊C4 in GL4(𝔽113) generated by

 1 0 0 0 0 112 0 0 0 0 106 0 0 0 0 106
,
 98 0 0 0 0 1 0 0 0 0 7 21 0 0 106 106
,
 15 0 0 0 0 1 0 0 0 0 1 2 0 0 0 112
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,106,0,0,0,0,106],[98,0,0,0,0,1,0,0,0,0,7,106,0,0,21,106],[15,0,0,0,0,1,0,0,0,0,1,0,0,0,2,112] >;

C14×C8⋊C4 in GAP, Magma, Sage, TeX

C_{14}\times C_8\rtimes C_4
% in TeX

G:=Group("C14xC8:C4");
// GroupNames label

G:=SmallGroup(448,811);
// by ID

G=gap.SmallGroup(448,811);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,3165,792,172]);
// Polycyclic

G:=Group<a,b,c|a^14=b^8=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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