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## G = (C4×Dic7)⋊8C4order 448 = 26·7

### 4th semidirect product of C4×Dic7 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — (C4×Dic7)⋊8C4
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×C4×Dic7 — (C4×Dic7)⋊8C4
 Lower central C7 — C2×C14 — (C4×Dic7)⋊8C4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C4×Dic7)⋊8C4
G = < a,b,c,d | a4=b14=d4=1, c2=b7, ab=ba, ac=ca, dad-1=a-1, cbc-1=b-1, bd=db, dcd-1=b7c >

Subgroups: 708 in 186 conjugacy classes, 95 normal (23 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×C4⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C429C4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C22×Dic7, C22×C28, C22×C28, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C14×C4⋊C4, (C4×Dic7)⋊8C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C41D4, C4⋊Q8, Dic14, C4×D7, C7⋊D4, C22×D7, C429C4, Dic7⋊C4, C2×Dic14, C2×C4×D7, D4×D7, Q8×D7, C2×C7⋊D4, C28⋊Q8, D7×C4⋊C4, C2×Dic7⋊C4, C28⋊D4, Dic7⋊Q8, (C4×Dic7)⋊8C4

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

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

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

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

88 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 7A 7B 7C 14A ··· 14U 28A ··· 28AJ order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 2 2 2 4 4 4 4 14 ··· 14 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + - + + - + - image C1 C2 C2 C2 C2 C4 D4 Q8 D4 Q8 D7 D14 Dic14 C4×D7 C7⋊D4 D4×D7 Q8×D7 kernel (C4×Dic7)⋊8C4 C2×C4×Dic7 C2×Dic7⋊C4 C2×C4⋊Dic7 C14×C4⋊C4 C4×Dic7 C2×Dic7 C2×Dic7 C2×C28 C2×C28 C2×C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C22 C22 # reps 1 1 4 1 1 8 4 4 2 2 3 9 12 12 12 6 6

Matrix representation of (C4×Dic7)⋊8C4 in GL5(𝔽29)

 1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 20 22 0 0 0 20 9
,
 1 0 0 0 0 0 22 1 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 28 0 0 0 0 0 12 0 0 0 0 26 17 0 0 0 0 0 1 0 0 0 0 0 1
,
 17 0 0 0 0 0 9 14 0 0 0 15 20 0 0 0 0 0 1 27 0 0 0 0 28

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,20,20,0,0,0,22,9],[1,0,0,0,0,0,22,28,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,12,26,0,0,0,0,17,0,0,0,0,0,1,0,0,0,0,0,1],[17,0,0,0,0,0,9,15,0,0,0,14,20,0,0,0,0,0,1,0,0,0,0,27,28] >;

(C4×Dic7)⋊8C4 in GAP, Magma, Sage, TeX

(C_4\times {\rm Dic}_7)\rtimes_8C_4
% in TeX

G:=Group("(C4xDic7):8C4");
// GroupNames label

G:=SmallGroup(448,510);
// by ID

G=gap.SmallGroup(448,510);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,422,387,58,18822]);
// 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,d*a*d^-1=a^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^7*c>;
// generators/relations

׿
×
𝔽