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## G = Dic7⋊C42order 448 = 26·7

### 1st semidirect product of Dic7 and C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic7⋊C42
G = < a,b,c,d | a14=c4=d4=1, b2=a7, bab-1=cac-1=a-1, ad=da, bc=cb, dbd-1=a7b, cd=dc >

Subgroups: 668 in 194 conjugacy classes, 99 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic7, Dic7, 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, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C14.C42, C7×C2.C42, C2×C4×Dic7, C2×C4×Dic7, C2×Dic7⋊C4, Dic7⋊C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C42, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, D14, C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×D7, C22×D7, C4×C4⋊C4, C2×C4×D7, D4×D7, D42D7, Q8×D7, D7×C42, C23.11D14, Dic74D4, Dic73Q8, D7×C4⋊C4, Dic7⋊C42

Smallest permutation representation of Dic7⋊C42
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 44 8 51)(2 43 9 50)(3 56 10 49)(4 55 11 48)(5 54 12 47)(6 53 13 46)(7 52 14 45)(15 314 22 321)(16 313 23 320)(17 312 24 319)(18 311 25 318)(19 310 26 317)(20 309 27 316)(21 322 28 315)(29 328 36 335)(30 327 37 334)(31 326 38 333)(32 325 39 332)(33 324 40 331)(34 323 41 330)(35 336 42 329)(57 92 64 85)(58 91 65 98)(59 90 66 97)(60 89 67 96)(61 88 68 95)(62 87 69 94)(63 86 70 93)(71 438 78 445)(72 437 79 444)(73 436 80 443)(74 435 81 442)(75 448 82 441)(76 447 83 440)(77 446 84 439)(99 348 106 341)(100 347 107 340)(101 346 108 339)(102 345 109 338)(103 344 110 337)(104 343 111 350)(105 342 112 349)(113 180 120 173)(114 179 121 172)(115 178 122 171)(116 177 123 170)(117 176 124 169)(118 175 125 182)(119 174 126 181)(127 413 134 420)(128 412 135 419)(129 411 136 418)(130 410 137 417)(131 409 138 416)(132 408 139 415)(133 407 140 414)(141 259 148 266)(142 258 149 265)(143 257 150 264)(144 256 151 263)(145 255 152 262)(146 254 153 261)(147 253 154 260)(155 401 162 394)(156 400 163 393)(157 399 164 406)(158 398 165 405)(159 397 166 404)(160 396 167 403)(161 395 168 402)(183 215 190 222)(184 214 191 221)(185 213 192 220)(186 212 193 219)(187 211 194 218)(188 224 195 217)(189 223 196 216)(197 359 204 352)(198 358 205 351)(199 357 206 364)(200 356 207 363)(201 355 208 362)(202 354 209 361)(203 353 210 360)(225 433 232 426)(226 432 233 425)(227 431 234 424)(228 430 235 423)(229 429 236 422)(230 428 237 421)(231 427 238 434)(239 298 246 305)(240 297 247 304)(241 296 248 303)(242 295 249 302)(243 308 250 301)(244 307 251 300)(245 306 252 299)(267 383 274 390)(268 382 275 389)(269 381 276 388)(270 380 277 387)(271 379 278 386)(272 392 279 385)(273 391 280 384)(281 375 288 368)(282 374 289 367)(283 373 290 366)(284 372 291 365)(285 371 292 378)(286 370 293 377)(287 369 294 376)
(1 71 340 399)(2 84 341 398)(3 83 342 397)(4 82 343 396)(5 81 344 395)(6 80 345 394)(7 79 346 393)(8 78 347 406)(9 77 348 405)(10 76 349 404)(11 75 350 403)(12 74 337 402)(13 73 338 401)(14 72 339 400)(15 34 122 408)(16 33 123 407)(17 32 124 420)(18 31 125 419)(19 30 126 418)(20 29 113 417)(21 42 114 416)(22 41 115 415)(23 40 116 414)(24 39 117 413)(25 38 118 412)(26 37 119 411)(27 36 120 410)(28 35 121 409)(43 439 99 165)(44 438 100 164)(45 437 101 163)(46 436 102 162)(47 435 103 161)(48 448 104 160)(49 447 105 159)(50 446 106 158)(51 445 107 157)(52 444 108 156)(53 443 109 155)(54 442 110 168)(55 441 111 167)(56 440 112 166)(57 213 230 291)(58 212 231 290)(59 211 232 289)(60 224 233 288)(61 223 234 287)(62 222 235 286)(63 221 236 285)(64 220 237 284)(65 219 238 283)(66 218 225 282)(67 217 226 281)(68 216 227 294)(69 215 228 293)(70 214 229 292)(85 185 421 372)(86 184 422 371)(87 183 423 370)(88 196 424 369)(89 195 425 368)(90 194 426 367)(91 193 427 366)(92 192 428 365)(93 191 429 378)(94 190 430 377)(95 189 431 376)(96 188 432 375)(97 187 433 374)(98 186 434 373)(127 312 325 169)(128 311 326 182)(129 310 327 181)(130 309 328 180)(131 322 329 179)(132 321 330 178)(133 320 331 177)(134 319 332 176)(135 318 333 175)(136 317 334 174)(137 316 335 173)(138 315 336 172)(139 314 323 171)(140 313 324 170)(141 299 272 360)(142 298 273 359)(143 297 274 358)(144 296 275 357)(145 295 276 356)(146 308 277 355)(147 307 278 354)(148 306 279 353)(149 305 280 352)(150 304 267 351)(151 303 268 364)(152 302 269 363)(153 301 270 362)(154 300 271 361)(197 265 239 384)(198 264 240 383)(199 263 241 382)(200 262 242 381)(201 261 243 380)(202 260 244 379)(203 259 245 392)(204 258 246 391)(205 257 247 390)(206 256 248 389)(207 255 249 388)(208 254 250 387)(209 253 251 386)(210 266 252 385)
(1 332 386 430)(2 333 387 431)(3 334 388 432)(4 335 389 433)(5 336 390 434)(6 323 391 421)(7 324 392 422)(8 325 379 423)(9 326 380 424)(10 327 381 425)(11 328 382 426)(12 329 383 427)(13 330 384 428)(14 331 385 429)(15 305 220 155)(16 306 221 156)(17 307 222 157)(18 308 223 158)(19 295 224 159)(20 296 211 160)(21 297 212 161)(22 298 213 162)(23 299 214 163)(24 300 215 164)(25 301 216 165)(26 302 217 166)(27 303 218 167)(28 304 219 168)(29 275 232 48)(30 276 233 49)(31 277 234 50)(32 278 235 51)(33 279 236 52)(34 280 237 53)(35 267 238 54)(36 268 225 55)(37 269 226 56)(38 270 227 43)(39 271 228 44)(40 272 229 45)(41 273 230 46)(42 274 231 47)(57 102 415 142)(58 103 416 143)(59 104 417 144)(60 105 418 145)(61 106 419 146)(62 107 420 147)(63 108 407 148)(64 109 408 149)(65 110 409 150)(66 111 410 151)(67 112 411 152)(68 99 412 153)(69 100 413 154)(70 101 414 141)(71 176 209 377)(72 177 210 378)(73 178 197 365)(74 179 198 366)(75 180 199 367)(76 181 200 368)(77 182 201 369)(78 169 202 370)(79 170 203 371)(80 171 204 372)(81 172 205 373)(82 173 206 374)(83 174 207 375)(84 175 208 376)(85 345 139 258)(86 346 140 259)(87 347 127 260)(88 348 128 261)(89 349 129 262)(90 350 130 263)(91 337 131 264)(92 338 132 265)(93 339 133 266)(94 340 134 253)(95 341 135 254)(96 342 136 255)(97 343 137 256)(98 344 138 257)(113 357 289 448)(114 358 290 435)(115 359 291 436)(116 360 292 437)(117 361 293 438)(118 362 294 439)(119 363 281 440)(120 364 282 441)(121 351 283 442)(122 352 284 443)(123 353 285 444)(124 354 286 445)(125 355 287 446)(126 356 288 447)(183 406 312 244)(184 393 313 245)(185 394 314 246)(186 395 315 247)(187 396 316 248)(188 397 317 249)(189 398 318 250)(190 399 319 251)(191 400 320 252)(192 401 321 239)(193 402 322 240)(194 403 309 241)(195 404 310 242)(196 405 311 243)

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

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

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

100 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 4U ··· 4AF 7A 7B 7C 14A ··· 14U 28A ··· 28AJ order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 ··· 2 7 ··· 7 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + - - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D7 C4○D4 D14 C4×D7 D4×D7 D4⋊2D7 Q8×D7 kernel Dic7⋊C42 C14.C42 C7×C2.C42 C2×C4×Dic7 C2×Dic7⋊C4 C4×Dic7 Dic7⋊C4 C2×Dic7 C2×Dic7 C2.C42 C2×C14 C22×C4 C2×C4 C22 C22 C22 # reps 1 1 1 3 2 8 16 2 2 3 4 9 36 3 6 3

Matrix representation of Dic7⋊C42 in GL6(𝔽29)

 0 1 0 0 0 0 28 7 0 0 0 0 0 0 28 1 0 0 0 0 9 19 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 4 9 0 0 0 0 8 25 0 0 0 0 0 0 28 1 0 0 0 0 0 1 0 0 0 0 0 0 27 28 0 0 0 0 5 2
,
 10 8 0 0 0 0 20 19 0 0 0 0 0 0 28 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 17 0 0 0 0 0 0 17 0 0 0 0 0 0 17 0 0 0 0 0 0 17 0 0 0 0 0 0 12 18 0 0 0 0 13 17

G:=sub<GL(6,GF(29))| [0,28,0,0,0,0,1,7,0,0,0,0,0,0,28,9,0,0,0,0,1,19,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[4,8,0,0,0,0,9,25,0,0,0,0,0,0,28,0,0,0,0,0,1,1,0,0,0,0,0,0,27,5,0,0,0,0,28,2],[10,20,0,0,0,0,8,19,0,0,0,0,0,0,28,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,12,13,0,0,0,0,18,17] >;

Dic7⋊C42 in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes C_4^2
% in TeX

G:=Group("Dic7:C4^2");
// GroupNames label

G:=SmallGroup(448,183);
// by ID

G=gap.SmallGroup(448,183);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,555,58,18822]);
// Polycyclic

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

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