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## G = C2×C28.44D4order 448 = 26·7

### Direct product of C2 and C28.44D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×C28.44D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4⋊Dic7 — C2×C4⋊Dic7 — C2×C28.44D4
 Lower central C7 — C14 — C28 — C2×C28.44D4
 Upper central C1 — C23 — C22×C4 — C22×C8

Generators and relations for C2×C28.44D4
G = < a,b,c,d | a2=b28=c4=1, d2=b14, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b21c-1 >

Subgroups: 740 in 162 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C14, C14, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C56, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, C2×Q8⋊C4, C4⋊Dic7, C4⋊Dic7, C2×C56, C2×C56, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, C28.44D4, C2×C4⋊Dic7, C22×C56, C22×Dic14, C2×C28.44D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, SD16, Q16, C22×C4, C2×D4, D14, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C4×D7, D28, C7⋊D4, C22×D7, C2×Q8⋊C4, C56⋊C2, Dic28, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, C28.44D4, C2×C56⋊C2, C2×Dic28, C2×D14⋊C4, C2×C28.44D4

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

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

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

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

124 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 7A 7B 7C 8A ··· 8H 14A ··· 14U 28A ··· 28X 56A ··· 56AV order 1 2 ··· 2 4 4 4 4 4 ··· 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 ··· 1 2 2 2 2 28 ··· 28 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + - + + + + - image C1 C2 C2 C2 C2 C4 D4 D4 D7 SD16 Q16 D14 D14 C4×D7 D28 C7⋊D4 D28 C56⋊C2 Dic28 kernel C2×C28.44D4 C28.44D4 C2×C4⋊Dic7 C22×C56 C22×Dic14 C2×Dic14 C2×C28 C22×C14 C22×C8 C2×C14 C2×C14 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 3 1 3 4 4 6 3 12 6 12 6 24 24

Matrix representation of C2×C28.44D4 in GL4(𝔽113) generated by

 1 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 112 0 0 0 0 1 0 0 0 0 19 100 0 0 13 9
,
 98 0 0 0 0 1 0 0 0 0 86 8 0 0 22 27
,
 112 0 0 0 0 1 0 0 0 0 55 77 0 0 15 58
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,1,0,0,0,0,19,13,0,0,100,9],[98,0,0,0,0,1,0,0,0,0,86,22,0,0,8,27],[112,0,0,0,0,1,0,0,0,0,55,15,0,0,77,58] >;

C2×C28.44D4 in GAP, Magma, Sage, TeX

C_2\times C_{28}._{44}D_4
% in TeX

G:=Group("C2xC28.44D4");
// GroupNames label

G:=SmallGroup(448,637);
// by ID

G=gap.SmallGroup(448,637);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,254,142,1123,136,18822]);
// 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=d*b*d^-1=b^-1,d*c*d^-1=b^21*c^-1>;
// generators/relations

׿
×
𝔽