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## G = C4⋊C4×C2×C14order 448 = 26·7

### Direct product of C2×C14 and C4⋊C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4⋊C4×C2×C14
 Chief series C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C4⋊C4 — C14×C4⋊C4 — C4⋊C4×C2×C14
 Lower central C1 — C2 — C4⋊C4×C2×C14
 Upper central C1 — C23×C14 — C4⋊C4×C2×C14

Generators and relations for C4⋊C4×C2×C14
G = < a,b,c,d | a2=b14=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 498 in 418 conjugacy classes, 338 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C22×C4, C22×C4, C24, C28, C28, C2×C14, C2×C14, C2×C4⋊C4, C23×C4, C23×C4, C2×C28, C2×C28, C22×C14, C22×C4⋊C4, C7×C4⋊C4, C22×C28, C22×C28, C23×C14, C14×C4⋊C4, C23×C28, C23×C28, C4⋊C4×C2×C14
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C28, C2×C14, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C4⋊C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C23×C14, C14×C4⋊C4, C23×C28, D4×C2×C14, Q8×C2×C14, C4⋊C4×C2×C14

Smallest permutation representation of C4⋊C4×C2×C14
Regular action on 448 points
Generators in S448
(1 371)(2 372)(3 373)(4 374)(5 375)(6 376)(7 377)(8 378)(9 365)(10 366)(11 367)(12 368)(13 369)(14 370)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 202)(30 203)(31 204)(32 205)(33 206)(34 207)(35 208)(36 209)(37 210)(38 197)(39 198)(40 199)(41 200)(42 201)(43 412)(44 413)(45 414)(46 415)(47 416)(48 417)(49 418)(50 419)(51 420)(52 407)(53 408)(54 409)(55 410)(56 411)(57 162)(58 163)(59 164)(60 165)(61 166)(62 167)(63 168)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 161)(71 352)(72 353)(73 354)(74 355)(75 356)(76 357)(77 358)(78 359)(79 360)(80 361)(81 362)(82 363)(83 364)(84 351)(85 279)(86 280)(87 267)(88 268)(89 269)(90 270)(91 271)(92 272)(93 273)(94 274)(95 275)(96 276)(97 277)(98 278)(113 254)(114 255)(115 256)(116 257)(117 258)(118 259)(119 260)(120 261)(121 262)(122 263)(123 264)(124 265)(125 266)(126 253)(127 211)(128 212)(129 213)(130 214)(131 215)(132 216)(133 217)(134 218)(135 219)(136 220)(137 221)(138 222)(139 223)(140 224)(141 193)(142 194)(143 195)(144 196)(145 183)(146 184)(147 185)(148 186)(149 187)(150 188)(151 189)(152 190)(153 191)(154 192)(169 248)(170 249)(171 250)(172 251)(173 252)(174 239)(175 240)(176 241)(177 242)(178 243)(179 244)(180 245)(181 246)(182 247)(225 422)(226 423)(227 424)(228 425)(229 426)(230 427)(231 428)(232 429)(233 430)(234 431)(235 432)(236 433)(237 434)(238 421)(281 350)(282 337)(283 338)(284 339)(285 340)(286 341)(287 342)(288 343)(289 344)(290 345)(291 346)(292 347)(293 348)(294 349)(295 322)(296 309)(297 310)(298 311)(299 312)(300 313)(301 314)(302 315)(303 316)(304 317)(305 318)(306 319)(307 320)(308 321)(323 391)(324 392)(325 379)(326 380)(327 381)(328 382)(329 383)(330 384)(331 385)(332 386)(333 387)(334 388)(335 389)(336 390)(393 440)(394 441)(395 442)(396 443)(397 444)(398 445)(399 446)(400 447)(401 448)(402 435)(403 436)(404 437)(405 438)(406 439)
(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)
(1 166 431 54)(2 167 432 55)(3 168 433 56)(4 155 434 43)(5 156 421 44)(6 157 422 45)(7 158 423 46)(8 159 424 47)(9 160 425 48)(10 161 426 49)(11 162 427 50)(12 163 428 51)(13 164 429 52)(14 165 430 53)(15 392 361 90)(16 379 362 91)(17 380 363 92)(18 381 364 93)(19 382 351 94)(20 383 352 95)(21 384 353 96)(22 385 354 97)(23 386 355 98)(24 387 356 85)(25 388 357 86)(26 389 358 87)(27 390 359 88)(28 391 360 89)(29 343 304 174)(30 344 305 175)(31 345 306 176)(32 346 307 177)(33 347 308 178)(34 348 295 179)(35 349 296 180)(36 350 297 181)(37 337 298 182)(38 338 299 169)(39 339 300 170)(40 340 301 171)(41 341 302 172)(42 342 303 173)(57 230 419 367)(58 231 420 368)(59 232 407 369)(60 233 408 370)(61 234 409 371)(62 235 410 372)(63 236 411 373)(64 237 412 374)(65 238 413 375)(66 225 414 376)(67 226 415 377)(68 227 416 378)(69 228 417 365)(70 229 418 366)(71 275 112 329)(72 276 99 330)(73 277 100 331)(74 278 101 332)(75 279 102 333)(76 280 103 334)(77 267 104 335)(78 268 105 336)(79 269 106 323)(80 270 107 324)(81 271 108 325)(82 272 109 326)(83 273 110 327)(84 274 111 328)(113 140 189 444)(114 127 190 445)(115 128 191 446)(116 129 192 447)(117 130 193 448)(118 131 194 435)(119 132 195 436)(120 133 196 437)(121 134 183 438)(122 135 184 439)(123 136 185 440)(124 137 186 441)(125 138 187 442)(126 139 188 443)(141 401 258 214)(142 402 259 215)(143 403 260 216)(144 404 261 217)(145 405 262 218)(146 406 263 219)(147 393 264 220)(148 394 265 221)(149 395 266 222)(150 396 253 223)(151 397 254 224)(152 398 255 211)(153 399 256 212)(154 400 257 213)(197 283 312 248)(198 284 313 249)(199 285 314 250)(200 286 315 251)(201 287 316 252)(202 288 317 239)(203 289 318 240)(204 290 319 241)(205 291 320 242)(206 292 321 243)(207 293 322 244)(208 294 309 245)(209 281 310 246)(210 282 311 247)
(1 363 445 36)(2 364 446 37)(3 351 447 38)(4 352 448 39)(5 353 435 40)(6 354 436 41)(7 355 437 42)(8 356 438 29)(9 357 439 30)(10 358 440 31)(11 359 441 32)(12 360 442 33)(13 361 443 34)(14 362 444 35)(15 139 295 429)(16 140 296 430)(17 127 297 431)(18 128 298 432)(19 129 299 433)(20 130 300 434)(21 131 301 421)(22 132 302 422)(23 133 303 423)(24 134 304 424)(25 135 305 425)(26 136 306 426)(27 137 307 427)(28 138 308 428)(43 95 193 339)(44 96 194 340)(45 97 195 341)(46 98 196 342)(47 85 183 343)(48 86 184 344)(49 87 185 345)(50 88 186 346)(51 89 187 347)(52 90 188 348)(53 91 189 349)(54 92 190 350)(55 93 191 337)(56 94 192 338)(57 336 265 242)(58 323 266 243)(59 324 253 244)(60 325 254 245)(61 326 255 246)(62 327 256 247)(63 328 257 248)(64 329 258 249)(65 330 259 250)(66 331 260 251)(67 332 261 252)(68 333 262 239)(69 334 263 240)(70 335 264 241)(71 401 198 374)(72 402 199 375)(73 403 200 376)(74 404 201 377)(75 405 202 378)(76 406 203 365)(77 393 204 366)(78 394 205 367)(79 395 206 368)(80 396 207 369)(81 397 208 370)(82 398 209 371)(83 399 210 372)(84 400 197 373)(99 215 314 238)(100 216 315 225)(101 217 316 226)(102 218 317 227)(103 219 318 228)(104 220 319 229)(105 221 320 230)(106 222 321 231)(107 223 322 232)(108 224 309 233)(109 211 310 234)(110 212 311 235)(111 213 312 236)(112 214 313 237)(113 180 165 379)(114 181 166 380)(115 182 167 381)(116 169 168 382)(117 170 155 383)(118 171 156 384)(119 172 157 385)(120 173 158 386)(121 174 159 387)(122 175 160 388)(123 176 161 389)(124 177 162 390)(125 178 163 391)(126 179 164 392)(141 284 412 275)(142 285 413 276)(143 286 414 277)(144 287 415 278)(145 288 416 279)(146 289 417 280)(147 290 418 267)(148 291 419 268)(149 292 420 269)(150 293 407 270)(151 294 408 271)(152 281 409 272)(153 282 410 273)(154 283 411 274)

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

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

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

280 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4X 7A ··· 7F 14A ··· 14CL 28A ··· 28EN order 1 2 ··· 2 4 ··· 4 7 ··· 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

280 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C4 C7 C14 C14 C28 D4 Q8 C7×D4 C7×Q8 kernel C4⋊C4×C2×C14 C14×C4⋊C4 C23×C28 C22×C28 C22×C4⋊C4 C2×C4⋊C4 C23×C4 C22×C4 C22×C14 C22×C14 C23 C23 # reps 1 12 3 16 6 72 18 96 4 4 24 24

Matrix representation of C4⋊C4×C2×C14 in GL5(𝔽29)

 28 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 28 0
,
 17 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 20 18 0 0 0 18 9

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,1,0],[17,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,20,18,0,0,0,18,9] >;

C4⋊C4×C2×C14 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\times C_2\times C_{14}
% in TeX

G:=Group("C4:C4xC2xC14");
// GroupNames label

G:=SmallGroup(448,1296);
// by ID

G=gap.SmallGroup(448,1296);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,792]);
// 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,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽