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## G = (C2×C28).288D4order 448 = 26·7

### 262nd non-split extension by C2×C28 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C14 — (C2×C28).288D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×Dic7⋊C4 — (C2×C28).288D4
 Lower central C7 — C22×C14 — (C2×C28).288D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C28).288D4
G = < a,b,c,d | a2=b28=c4=1, d2=b14, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=ab13, dcd-1=ac-1 >

Subgroups: 548 in 134 conjugacy classes, 57 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.83C23, Dic7⋊C4, C7×C4⋊C4, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C14.C42, C14.C42, C2×Dic7⋊C4, C14×C4⋊C4, (C2×C28).288D4
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C7⋊D4, C22×D7, C23.83C23, C4○D28, D42D7, Q8×D7, Q82D7, C2×C7⋊D4, Dic7.Q8, C4⋊C4⋊D7, C23.23D14, C28.17D4, D143Q8, (C2×C28).288D4

Smallest permutation representation of (C2×C28).288D4
Regular action on 448 points
Generators in S448
(1 268)(2 269)(3 270)(4 271)(5 272)(6 273)(7 274)(8 275)(9 276)(10 277)(11 278)(12 279)(13 280)(14 253)(15 254)(16 255)(17 256)(18 257)(19 258)(20 259)(21 260)(22 261)(23 262)(24 263)(25 264)(26 265)(27 266)(28 267)(29 171)(30 172)(31 173)(32 174)(33 175)(34 176)(35 177)(36 178)(37 179)(38 180)(39 181)(40 182)(41 183)(42 184)(43 185)(44 186)(45 187)(46 188)(47 189)(48 190)(49 191)(50 192)(51 193)(52 194)(53 195)(54 196)(55 169)(56 170)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(113 341)(114 342)(115 343)(116 344)(117 345)(118 346)(119 347)(120 348)(121 349)(122 350)(123 351)(124 352)(125 353)(126 354)(127 355)(128 356)(129 357)(130 358)(131 359)(132 360)(133 361)(134 362)(135 363)(136 364)(137 337)(138 338)(139 339)(140 340)(141 383)(142 384)(143 385)(144 386)(145 387)(146 388)(147 389)(148 390)(149 391)(150 392)(151 365)(152 366)(153 367)(154 368)(155 369)(156 370)(157 371)(158 372)(159 373)(160 374)(161 375)(162 376)(163 377)(164 378)(165 379)(166 380)(167 381)(168 382)(197 304)(198 305)(199 306)(200 307)(201 308)(202 281)(203 282)(204 283)(205 284)(206 285)(207 286)(208 287)(209 288)(210 289)(211 290)(212 291)(213 292)(214 293)(215 294)(216 295)(217 296)(218 297)(219 298)(220 299)(221 300)(222 301)(223 302)(224 303)(225 332)(226 333)(227 334)(228 335)(229 336)(230 309)(231 310)(232 311)(233 312)(234 313)(235 314)(236 315)(237 316)(238 317)(239 318)(240 319)(241 320)(242 321)(243 322)(244 323)(245 324)(246 325)(247 326)(248 327)(249 328)(250 329)(251 330)(252 331)(393 426)(394 427)(395 428)(396 429)(397 430)(398 431)(399 432)(400 433)(401 434)(402 435)(403 436)(404 437)(405 438)(406 439)(407 440)(408 441)(409 442)(410 443)(411 444)(412 445)(413 446)(414 447)(415 448)(416 421)(417 422)(418 423)(419 424)(420 425)
(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 53 235 291)(2 194 236 211)(3 51 237 289)(4 192 238 209)(5 49 239 287)(6 190 240 207)(7 47 241 285)(8 188 242 205)(9 45 243 283)(10 186 244 203)(11 43 245 281)(12 184 246 201)(13 41 247 307)(14 182 248 199)(15 39 249 305)(16 180 250 197)(17 37 251 303)(18 178 252 223)(19 35 225 301)(20 176 226 221)(21 33 227 299)(22 174 228 219)(23 31 229 297)(24 172 230 217)(25 29 231 295)(26 170 232 215)(27 55 233 293)(28 196 234 213)(30 309 296 263)(32 335 298 261)(34 333 300 259)(36 331 302 257)(38 329 304 255)(40 327 306 253)(42 325 308 279)(44 323 282 277)(46 321 284 275)(48 319 286 273)(50 317 288 271)(52 315 290 269)(54 313 292 267)(56 311 294 265)(57 403 339 372)(58 435 340 157)(59 401 341 370)(60 433 342 155)(61 399 343 368)(62 431 344 153)(63 397 345 366)(64 429 346 151)(65 395 347 392)(66 427 348 149)(67 393 349 390)(68 425 350 147)(69 419 351 388)(70 423 352 145)(71 417 353 386)(72 421 354 143)(73 415 355 384)(74 447 356 141)(75 413 357 382)(76 445 358 167)(77 411 359 380)(78 443 360 165)(79 409 361 378)(80 441 362 163)(81 407 363 376)(82 439 364 161)(83 405 337 374)(84 437 338 159)(85 416 126 385)(86 448 127 142)(87 414 128 383)(88 446 129 168)(89 412 130 381)(90 444 131 166)(91 410 132 379)(92 442 133 164)(93 408 134 377)(94 440 135 162)(95 406 136 375)(96 438 137 160)(97 404 138 373)(98 436 139 158)(99 402 140 371)(100 434 113 156)(101 400 114 369)(102 432 115 154)(103 398 116 367)(104 430 117 152)(105 396 118 365)(106 428 119 150)(107 394 120 391)(108 426 121 148)(109 420 122 389)(110 424 123 146)(111 418 124 387)(112 422 125 144)(169 312 214 266)(171 310 216 264)(173 336 218 262)(175 334 220 260)(177 332 222 258)(179 330 224 256)(181 328 198 254)(183 326 200 280)(185 324 202 278)(187 322 204 276)(189 320 206 274)(191 318 208 272)(193 316 210 270)(195 314 212 268)
(1 60 15 74)(2 86 16 100)(3 58 17 72)(4 112 18 98)(5 84 19 70)(6 110 20 96)(7 82 21 68)(8 108 22 94)(9 80 23 66)(10 106 24 92)(11 78 25 64)(12 104 26 90)(13 76 27 62)(14 102 28 88)(29 365 43 379)(30 164 44 150)(31 391 45 377)(32 162 46 148)(33 389 47 375)(34 160 48 146)(35 387 49 373)(36 158 50 144)(37 385 51 371)(38 156 52 142)(39 383 53 369)(40 154 54 168)(41 381 55 367)(42 152 56 166)(57 271 71 257)(59 269 73 255)(61 267 75 253)(63 265 77 279)(65 263 79 277)(67 261 81 275)(69 259 83 273)(85 270 99 256)(87 268 101 254)(89 266 103 280)(91 264 105 278)(93 262 107 276)(95 260 109 274)(97 258 111 272)(113 236 127 250)(114 328 128 314)(115 234 129 248)(116 326 130 312)(117 232 131 246)(118 324 132 310)(119 230 133 244)(120 322 134 336)(121 228 135 242)(122 320 136 334)(123 226 137 240)(124 318 138 332)(125 252 139 238)(126 316 140 330)(141 195 155 181)(143 193 157 179)(145 191 159 177)(147 189 161 175)(149 187 163 173)(151 185 165 171)(153 183 167 169)(170 380 184 366)(172 378 186 392)(174 376 188 390)(176 374 190 388)(178 372 192 386)(180 370 194 384)(182 368 196 382)(197 401 211 415)(198 447 212 433)(199 399 213 413)(200 445 214 431)(201 397 215 411)(202 443 216 429)(203 395 217 409)(204 441 218 427)(205 393 219 407)(206 439 220 425)(207 419 221 405)(208 437 222 423)(209 417 223 403)(210 435 224 421)(225 352 239 338)(227 350 241 364)(229 348 243 362)(231 346 245 360)(233 344 247 358)(235 342 249 356)(237 340 251 354)(281 410 295 396)(282 428 296 442)(283 408 297 394)(284 426 298 440)(285 406 299 420)(286 424 300 438)(287 404 301 418)(288 422 302 436)(289 402 303 416)(290 448 304 434)(291 400 305 414)(292 446 306 432)(293 398 307 412)(294 444 308 430)(309 361 323 347)(311 359 325 345)(313 357 327 343)(315 355 329 341)(317 353 331 339)(319 351 333 337)(321 349 335 363)

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

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

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

82 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4F 4G ··· 4N 7A 7B 7C 14A ··· 14U 28A ··· 28AJ order 1 2 ··· 2 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 4 ··· 4 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + - + + + - - + image C1 C2 C2 C2 Q8 D4 D7 C4○D4 D14 C7⋊D4 C4○D28 D4⋊2D7 Q8×D7 Q8⋊2D7 kernel (C2×C28).288D4 C14.C42 C2×Dic7⋊C4 C14×C4⋊C4 C2×Dic7 C2×C28 C2×C4⋊C4 C2×C14 C22×C4 C2×C4 C22 C22 C22 C22 # reps 1 5 1 1 2 2 3 10 9 12 24 6 3 3

Matrix representation of (C2×C28).288D4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 24 16 0 0 0 0 13 5 0 0 0 0 0 0 5 27 0 0 0 0 2 28 0 0 0 0 0 0 26 3 0 0 0 0 16 3
,
 24 27 0 0 0 0 13 5 0 0 0 0 0 0 28 0 0 0 0 0 26 1 0 0 0 0 0 0 3 15 0 0 0 0 13 26
,
 27 5 0 0 0 0 11 2 0 0 0 0 0 0 27 11 0 0 0 0 5 2 0 0 0 0 0 0 12 0 0 0 0 0 0 12

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[24,13,0,0,0,0,16,5,0,0,0,0,0,0,5,2,0,0,0,0,27,28,0,0,0,0,0,0,26,16,0,0,0,0,3,3],[24,13,0,0,0,0,27,5,0,0,0,0,0,0,28,26,0,0,0,0,0,1,0,0,0,0,0,0,3,13,0,0,0,0,15,26],[27,11,0,0,0,0,5,2,0,0,0,0,0,0,27,5,0,0,0,0,11,2,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;

(C2×C28).288D4 in GAP, Magma, Sage, TeX

(C_2\times C_{28})._{288}D_4
% in TeX

G:=Group("(C2xC28).288D4");
// GroupNames label

G:=SmallGroup(448,516);
// by ID

G=gap.SmallGroup(448,516);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,232,254,387,268,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=c^4=1,d^2=b^14,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d^-1=a*b^13,d*c*d^-1=a*c^-1>;
// generators/relations

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