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## G = C4×Dic7⋊C4order 448 = 26·7

### Direct product of C4 and Dic7⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C4×Dic7⋊C4
G = < a,b,c,d | a4=b14=d4=1, c2=b7, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b7c >

Subgroups: 644 in 194 conjugacy classes, 107 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic7, Dic7, C28, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C4×C4⋊C4, C4×Dic7, C4×Dic7, Dic7⋊C4, C4×C28, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C14.C42, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4×C28, C4×Dic7⋊C4
Quotients:

Smallest permutation representation of C4×Dic7⋊C4
Regular action on 448 points
Generators in S448
(1 134 22 120)(2 135 23 121)(3 136 24 122)(4 137 25 123)(5 138 26 124)(6 139 27 125)(7 140 28 126)(8 127 15 113)(9 128 16 114)(10 129 17 115)(11 130 18 116)(12 131 19 117)(13 132 20 118)(14 133 21 119)(29 162 50 141)(30 163 51 142)(31 164 52 143)(32 165 53 144)(33 166 54 145)(34 167 55 146)(35 168 56 147)(36 155 43 148)(37 156 44 149)(38 157 45 150)(39 158 46 151)(40 159 47 152)(41 160 48 153)(42 161 49 154)(57 183 71 176)(58 184 72 177)(59 185 73 178)(60 186 74 179)(61 187 75 180)(62 188 76 181)(63 189 77 182)(64 190 78 169)(65 191 79 170)(66 192 80 171)(67 193 81 172)(68 194 82 173)(69 195 83 174)(70 196 84 175)(85 218 106 197)(86 219 107 198)(87 220 108 199)(88 221 109 200)(89 222 110 201)(90 223 111 202)(91 224 112 203)(92 211 99 204)(93 212 100 205)(94 213 101 206)(95 214 102 207)(96 215 103 208)(97 216 104 209)(98 217 105 210)(225 362 249 339)(226 363 250 340)(227 364 251 341)(228 351 252 342)(229 352 239 343)(230 353 240 344)(231 354 241 345)(232 355 242 346)(233 356 243 347)(234 357 244 348)(235 358 245 349)(236 359 246 350)(237 360 247 337)(238 361 248 338)(253 386 268 367)(254 387 269 368)(255 388 270 369)(256 389 271 370)(257 390 272 371)(258 391 273 372)(259 392 274 373)(260 379 275 374)(261 380 276 375)(262 381 277 376)(263 382 278 377)(264 383 279 378)(265 384 280 365)(266 385 267 366)(281 418 295 405)(282 419 296 406)(283 420 297 393)(284 407 298 394)(285 408 299 395)(286 409 300 396)(287 410 301 397)(288 411 302 398)(289 412 303 399)(290 413 304 400)(291 414 305 401)(292 415 306 402)(293 416 307 403)(294 417 308 404)(309 448 328 423)(310 435 329 424)(311 436 330 425)(312 437 331 426)(313 438 332 427)(314 439 333 428)(315 440 334 429)(316 441 335 430)(317 442 336 431)(318 443 323 432)(319 444 324 433)(320 445 325 434)(321 446 326 421)(322 447 327 422)
(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)
(1 263 8 256)(2 262 9 255)(3 261 10 254)(4 260 11 253)(5 259 12 266)(6 258 13 265)(7 257 14 264)(15 271 22 278)(16 270 23 277)(17 269 24 276)(18 268 25 275)(19 267 26 274)(20 280 27 273)(21 279 28 272)(29 226 36 233)(30 225 37 232)(31 238 38 231)(32 237 39 230)(33 236 40 229)(34 235 41 228)(35 234 42 227)(43 243 50 250)(44 242 51 249)(45 241 52 248)(46 240 53 247)(47 239 54 246)(48 252 55 245)(49 251 56 244)(57 319 64 312)(58 318 65 311)(59 317 66 310)(60 316 67 309)(61 315 68 322)(62 314 69 321)(63 313 70 320)(71 324 78 331)(72 323 79 330)(73 336 80 329)(74 335 81 328)(75 334 82 327)(76 333 83 326)(77 332 84 325)(85 290 92 283)(86 289 93 282)(87 288 94 281)(88 287 95 294)(89 286 96 293)(90 285 97 292)(91 284 98 291)(99 297 106 304)(100 296 107 303)(101 295 108 302)(102 308 109 301)(103 307 110 300)(104 306 111 299)(105 305 112 298)(113 370 120 377)(114 369 121 376)(115 368 122 375)(116 367 123 374)(117 366 124 373)(118 365 125 372)(119 378 126 371)(127 389 134 382)(128 388 135 381)(129 387 136 380)(130 386 137 379)(131 385 138 392)(132 384 139 391)(133 383 140 390)(141 340 148 347)(142 339 149 346)(143 338 150 345)(144 337 151 344)(145 350 152 343)(146 349 153 342)(147 348 154 341)(155 356 162 363)(156 355 163 362)(157 354 164 361)(158 353 165 360)(159 352 166 359)(160 351 167 358)(161 364 168 357)(169 426 176 433)(170 425 177 432)(171 424 178 431)(172 423 179 430)(173 422 180 429)(174 421 181 428)(175 434 182 427)(183 444 190 437)(184 443 191 436)(185 442 192 435)(186 441 193 448)(187 440 194 447)(188 439 195 446)(189 438 196 445)(197 400 204 393)(198 399 205 406)(199 398 206 405)(200 397 207 404)(201 396 208 403)(202 395 209 402)(203 394 210 401)(211 420 218 413)(212 419 219 412)(213 418 220 411)(214 417 221 410)(215 416 222 409)(216 415 223 408)(217 414 224 407)
(1 218 50 169)(2 219 51 170)(3 220 52 171)(4 221 53 172)(5 222 54 173)(6 223 55 174)(7 224 56 175)(8 211 43 176)(9 212 44 177)(10 213 45 178)(11 214 46 179)(12 215 47 180)(13 216 48 181)(14 217 49 182)(15 204 36 183)(16 205 37 184)(17 206 38 185)(18 207 39 186)(19 208 40 187)(20 209 41 188)(21 210 42 189)(22 197 29 190)(23 198 30 191)(24 199 31 192)(25 200 32 193)(26 201 33 194)(27 202 34 195)(28 203 35 196)(57 127 99 148)(58 128 100 149)(59 129 101 150)(60 130 102 151)(61 131 103 152)(62 132 104 153)(63 133 105 154)(64 134 106 141)(65 135 107 142)(66 136 108 143)(67 137 109 144)(68 138 110 145)(69 139 111 146)(70 140 112 147)(71 113 92 155)(72 114 93 156)(73 115 94 157)(74 116 95 158)(75 117 96 159)(76 118 97 160)(77 119 98 161)(78 120 85 162)(79 121 86 163)(80 122 87 164)(81 123 88 165)(82 124 89 166)(83 125 90 167)(84 126 91 168)(225 443 277 406)(226 444 278 393)(227 445 279 394)(228 446 280 395)(229 447 267 396)(230 448 268 397)(231 435 269 398)(232 436 270 399)(233 437 271 400)(234 438 272 401)(235 439 273 402)(236 440 274 403)(237 441 275 404)(238 442 276 405)(239 422 266 409)(240 423 253 410)(241 424 254 411)(242 425 255 412)(243 426 256 413)(244 427 257 414)(245 428 258 415)(246 429 259 416)(247 430 260 417)(248 431 261 418)(249 432 262 419)(250 433 263 420)(251 434 264 407)(252 421 265 408)(281 361 336 375)(282 362 323 376)(283 363 324 377)(284 364 325 378)(285 351 326 365)(286 352 327 366)(287 353 328 367)(288 354 329 368)(289 355 330 369)(290 356 331 370)(291 357 332 371)(292 358 333 372)(293 359 334 373)(294 360 335 374)(295 338 317 380)(296 339 318 381)(297 340 319 382)(298 341 320 383)(299 342 321 384)(300 343 322 385)(301 344 309 386)(302 345 310 387)(303 346 311 388)(304 347 312 389)(305 348 313 390)(306 349 314 391)(307 350 315 392)(308 337 316 379)

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

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

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

136 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 4Q ··· 4AF 7A 7B 7C 14A ··· 14U 28A ··· 28BT order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 1 ··· 1 2 ··· 2 14 ··· 14 2 2 2 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + - + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D7 C4○D4 D14 Dic14 C4×D7 C7⋊D4 C4○D28 kernel C4×Dic7⋊C4 C14.C42 C2×C4×Dic7 C2×Dic7⋊C4 C2×C4×C28 C4×Dic7 Dic7⋊C4 C2×C28 C2×C28 C2×C42 C2×C14 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 2 2 1 8 16 2 2 3 4 9 12 36 12 24

Matrix representation of C4×Dic7⋊C4 in GL6(𝔽29)

 12 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 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 4 28 0 0 0 0 5 28
,
 28 0 0 0 0 0 0 1 0 0 0 0 0 0 16 2 0 0 0 0 2 13 0 0 0 0 0 0 7 5 0 0 0 0 2 22
,
 28 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 28 0 0 0 0 0 0 0 28 0 0 0 0 0 0 28

G:=sub<GL(6,GF(29))| [12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,4,5,0,0,0,0,28,28],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,16,2,0,0,0,0,2,13,0,0,0,0,0,0,7,2,0,0,0,0,5,22],[28,0,0,0,0,0,0,12,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;

C4×Dic7⋊C4 in GAP, Magma, Sage, TeX

C_4\times {\rm Dic}_7\rtimes C_4
% in TeX

G:=Group("C4xDic7:C4");
// GroupNames label

G:=SmallGroup(448,465);
// by ID

G=gap.SmallGroup(448,465);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,758,58,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^14=d^4=1,c^2=b^7,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

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