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## G = C42⋊9Dic7order 448 = 26·7

### 6th semidirect product of C42 and Dic7 acting via Dic7/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C42⋊9Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C14.C42 — C42⋊9Dic7
 Lower central C7 — C2×C14 — C42⋊9Dic7
 Upper central C1 — C23 — C2×C42

Generators and relations for C429Dic7
G = < a,b,c,d | a4=b4=c14=1, d2=c7, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 580 in 154 conjugacy classes, 87 normal (17 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, C28, C28, C2×C14, C2×C14, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic7, C2×C28, C2×C28, C22×C14, C428C4, C4⋊Dic7, C4×C28, C22×Dic7, C22×C28, C22×C28, C14.C42, C2×C4⋊Dic7, C2×C4×C28, C429Dic7
Quotients:

Smallest permutation representation of C429Dic7
Regular action on 448 points
Generators in S448
(1 305 177 143)(2 306 178 144)(3 307 179 145)(4 308 180 146)(5 295 181 147)(6 296 182 148)(7 297 169 149)(8 298 170 150)(9 299 171 151)(10 300 172 152)(11 301 173 153)(12 302 174 154)(13 303 175 141)(14 304 176 142)(15 423 284 412)(16 424 285 413)(17 425 286 414)(18 426 287 415)(19 427 288 416)(20 428 289 417)(21 429 290 418)(22 430 291 419)(23 431 292 420)(24 432 293 407)(25 433 294 408)(26 434 281 409)(27 421 282 410)(28 422 283 411)(29 438 86 225)(30 439 87 226)(31 440 88 227)(32 441 89 228)(33 442 90 229)(34 443 91 230)(35 444 92 231)(36 445 93 232)(37 446 94 233)(38 447 95 234)(39 448 96 235)(40 435 97 236)(41 436 98 237)(42 437 85 238)(43 383 309 345)(44 384 310 346)(45 385 311 347)(46 386 312 348)(47 387 313 349)(48 388 314 350)(49 389 315 337)(50 390 316 338)(51 391 317 339)(52 392 318 340)(53 379 319 341)(54 380 320 342)(55 381 321 343)(56 382 322 344)(57 360 133 376)(58 361 134 377)(59 362 135 378)(60 363 136 365)(61 364 137 366)(62 351 138 367)(63 352 139 368)(64 353 140 369)(65 354 127 370)(66 355 128 371)(67 356 129 372)(68 357 130 373)(69 358 131 374)(70 359 132 375)(71 265 400 204)(72 266 401 205)(73 253 402 206)(74 254 403 207)(75 255 404 208)(76 256 405 209)(77 257 406 210)(78 258 393 197)(79 259 394 198)(80 260 395 199)(81 261 396 200)(82 262 397 201)(83 263 398 202)(84 264 399 203)(99 324 274 194)(100 325 275 195)(101 326 276 196)(102 327 277 183)(103 328 278 184)(104 329 279 185)(105 330 280 186)(106 331 267 187)(107 332 268 188)(108 333 269 189)(109 334 270 190)(110 335 271 191)(111 336 272 192)(112 323 273 193)(113 157 219 242)(114 158 220 243)(115 159 221 244)(116 160 222 245)(117 161 223 246)(118 162 224 247)(119 163 211 248)(120 164 212 249)(121 165 213 250)(122 166 214 251)(123 167 215 252)(124 168 216 239)(125 155 217 240)(126 156 218 241)
(1 382 28 96)(2 383 15 97)(3 384 16 98)(4 385 17 85)(5 386 18 86)(6 387 19 87)(7 388 20 88)(8 389 21 89)(9 390 22 90)(10 391 23 91)(11 392 24 92)(12 379 25 93)(13 380 26 94)(14 381 27 95)(29 181 348 287)(30 182 349 288)(31 169 350 289)(32 170 337 290)(33 171 338 291)(34 172 339 292)(35 173 340 293)(36 174 341 294)(37 175 342 281)(38 176 343 282)(39 177 344 283)(40 178 345 284)(41 179 346 285)(42 180 347 286)(43 412 435 144)(44 413 436 145)(45 414 437 146)(46 415 438 147)(47 416 439 148)(48 417 440 149)(49 418 441 150)(50 419 442 151)(51 420 443 152)(52 407 444 153)(53 408 445 154)(54 409 446 141)(55 410 447 142)(56 411 448 143)(57 77 106 242)(58 78 107 243)(59 79 108 244)(60 80 109 245)(61 81 110 246)(62 82 111 247)(63 83 112 248)(64 84 99 249)(65 71 100 250)(66 72 101 251)(67 73 102 252)(68 74 103 239)(69 75 104 240)(70 76 105 241)(113 360 257 331)(114 361 258 332)(115 362 259 333)(116 363 260 334)(117 364 261 335)(118 351 262 336)(119 352 263 323)(120 353 264 324)(121 354 265 325)(122 355 266 326)(123 356 253 327)(124 357 254 328)(125 358 255 329)(126 359 256 330)(127 400 275 165)(128 401 276 166)(129 402 277 167)(130 403 278 168)(131 404 279 155)(132 405 280 156)(133 406 267 157)(134 393 268 158)(135 394 269 159)(136 395 270 160)(137 396 271 161)(138 397 272 162)(139 398 273 163)(140 399 274 164)(183 215 372 206)(184 216 373 207)(185 217 374 208)(186 218 375 209)(187 219 376 210)(188 220 377 197)(189 221 378 198)(190 222 365 199)(191 223 366 200)(192 224 367 201)(193 211 368 202)(194 212 369 203)(195 213 370 204)(196 214 371 205)(225 295 312 426)(226 296 313 427)(227 297 314 428)(228 298 315 429)(229 299 316 430)(230 300 317 431)(231 301 318 432)(232 302 319 433)(233 303 320 434)(234 304 321 421)(235 305 322 422)(236 306 309 423)(237 307 310 424)(238 308 311 425)
(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)
(1 401 8 394)(2 400 9 393)(3 399 10 406)(4 398 11 405)(5 397 12 404)(6 396 13 403)(7 395 14 402)(15 165 22 158)(16 164 23 157)(17 163 24 156)(18 162 25 155)(19 161 26 168)(20 160 27 167)(21 159 28 166)(29 138 36 131)(30 137 37 130)(31 136 38 129)(32 135 39 128)(33 134 40 127)(34 133 41 140)(35 132 42 139)(43 370 50 377)(44 369 51 376)(45 368 52 375)(46 367 53 374)(47 366 54 373)(48 365 55 372)(49 378 56 371)(57 98 64 91)(58 97 65 90)(59 96 66 89)(60 95 67 88)(61 94 68 87)(62 93 69 86)(63 92 70 85)(71 171 78 178)(72 170 79 177)(73 169 80 176)(74 182 81 175)(75 181 82 174)(76 180 83 173)(77 179 84 172)(99 391 106 384)(100 390 107 383)(101 389 108 382)(102 388 109 381)(103 387 110 380)(104 386 111 379)(105 385 112 392)(113 145 120 152)(114 144 121 151)(115 143 122 150)(116 142 123 149)(117 141 124 148)(118 154 125 147)(119 153 126 146)(183 440 190 447)(184 439 191 446)(185 438 192 445)(186 437 193 444)(187 436 194 443)(188 435 195 442)(189 448 196 441)(197 423 204 430)(198 422 205 429)(199 421 206 428)(200 434 207 427)(201 433 208 426)(202 432 209 425)(203 431 210 424)(211 301 218 308)(212 300 219 307)(213 299 220 306)(214 298 221 305)(215 297 222 304)(216 296 223 303)(217 295 224 302)(225 336 232 329)(226 335 233 328)(227 334 234 327)(228 333 235 326)(229 332 236 325)(230 331 237 324)(231 330 238 323)(239 288 246 281)(240 287 247 294)(241 286 248 293)(242 285 249 292)(243 284 250 291)(244 283 251 290)(245 282 252 289)(253 417 260 410)(254 416 261 409)(255 415 262 408)(256 414 263 407)(257 413 264 420)(258 412 265 419)(259 411 266 418)(267 346 274 339)(268 345 275 338)(269 344 276 337)(270 343 277 350)(271 342 278 349)(272 341 279 348)(273 340 280 347)(309 354 316 361)(310 353 317 360)(311 352 318 359)(312 351 319 358)(313 364 320 357)(314 363 321 356)(315 362 322 355)

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

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

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

124 conjugacy classes

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

124 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + - + - + - + image C1 C2 C2 C2 C4 D4 Q8 D7 C4○D4 Dic7 D14 Dic14 D28 C4○D28 kernel C42⋊9Dic7 C14.C42 C2×C4⋊Dic7 C2×C4×C28 C4×C28 C2×C28 C2×C28 C2×C42 C2×C14 C42 C22×C4 C2×C4 C2×C4 C22 # reps 1 4 2 1 8 2 2 3 8 12 9 12 12 48

Matrix representation of C429Dic7 in GL5(𝔽29)

 28 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12
,
 28 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 24 14 0 0 0 19 5
,
 28 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 28 0 0 0 9 21
,
 17 0 0 0 0 0 23 21 0 0 0 8 6 0 0 0 0 0 2 8 0 0 0 3 27

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12],[28,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,24,19,0,0,0,14,5],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,9,0,0,0,28,21],[17,0,0,0,0,0,23,8,0,0,0,21,6,0,0,0,0,0,2,3,0,0,0,8,27] >;

C429Dic7 in GAP, Magma, Sage, TeX

C_4^2\rtimes_9{\rm Dic}_7
% in TeX

G:=Group("C4^2:9Dic7");
// GroupNames label

G:=SmallGroup(448,470);
// by ID

G=gap.SmallGroup(448,470);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,120,422,184,18822]);
// Polycyclic

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

׿
×
𝔽