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## G = C2×C4×C7⋊C8order 448 = 26·7

### Direct product of C2×C4 and C7⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×C4×C7⋊C8
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C7⋊C8 — C22×C7⋊C8 — C2×C4×C7⋊C8
 Lower central C7 — C2×C4×C7⋊C8
 Upper central C1 — C2×C42

Generators and relations for C2×C4×C7⋊C8
G = < a,b,c,d | a2=b4=c7=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 324 in 162 conjugacy classes, 135 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C22×C4, C22×C4, C28, C2×C14, C2×C14, C4×C8, C2×C42, C22×C8, C7⋊C8, C2×C28, C2×C28, C22×C14, C2×C4×C8, C2×C7⋊C8, C4×C28, C22×C28, C22×C28, C4×C7⋊C8, C22×C7⋊C8, C2×C4×C28, C2×C4×C7⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C42, C2×C8, C22×C4, Dic7, D14, C4×C8, C2×C42, C22×C8, C7⋊C8, C4×D7, C2×Dic7, C22×D7, C2×C4×C8, C2×C7⋊C8, C4×Dic7, C2×C4×D7, C22×Dic7, C4×C7⋊C8, C22×C7⋊C8, C2×C4×Dic7, C2×C4×C7⋊C8

Smallest permutation representation of C2×C4×C7⋊C8
Regular action on 448 points
Generators in S448
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 152)(10 145)(11 146)(12 147)(13 148)(14 149)(15 150)(16 151)(17 287)(18 288)(19 281)(20 282)(21 283)(22 284)(23 285)(24 286)(25 159)(26 160)(27 153)(28 154)(29 155)(30 156)(31 157)(32 158)(33 139)(34 140)(35 141)(36 142)(37 143)(38 144)(39 137)(40 138)(41 300)(42 301)(43 302)(44 303)(45 304)(46 297)(47 298)(48 299)(49 308)(50 309)(51 310)(52 311)(53 312)(54 305)(55 306)(56 307)(57 199)(58 200)(59 193)(60 194)(61 195)(62 196)(63 197)(64 198)(65 241)(66 242)(67 243)(68 244)(69 245)(70 246)(71 247)(72 248)(73 236)(74 237)(75 238)(76 239)(77 240)(78 233)(79 234)(80 235)(81 271)(82 272)(83 265)(84 266)(85 267)(86 268)(87 269)(88 270)(89 215)(90 216)(91 209)(92 210)(93 211)(94 212)(95 213)(96 214)(105 260)(106 261)(107 262)(108 263)(109 264)(110 257)(111 258)(112 259)(113 408)(114 401)(115 402)(116 403)(117 404)(118 405)(119 406)(120 407)(121 416)(122 409)(123 410)(124 411)(125 412)(126 413)(127 414)(128 415)(129 436)(130 437)(131 438)(132 439)(133 440)(134 433)(135 434)(136 435)(161 421)(162 422)(163 423)(164 424)(165 417)(166 418)(167 419)(168 420)(169 369)(170 370)(171 371)(172 372)(173 373)(174 374)(175 375)(176 376)(177 355)(178 356)(179 357)(180 358)(181 359)(182 360)(183 353)(184 354)(185 379)(186 380)(187 381)(188 382)(189 383)(190 384)(191 377)(192 378)(201 348)(202 349)(203 350)(204 351)(205 352)(206 345)(207 346)(208 347)(217 363)(218 364)(219 365)(220 366)(221 367)(222 368)(223 361)(224 362)(225 332)(226 333)(227 334)(228 335)(229 336)(230 329)(231 330)(232 331)(249 339)(250 340)(251 341)(252 342)(253 343)(254 344)(255 337)(256 338)(273 446)(274 447)(275 448)(276 441)(277 442)(278 443)(279 444)(280 445)(289 389)(290 390)(291 391)(292 392)(293 385)(294 386)(295 387)(296 388)(313 426)(314 427)(315 428)(316 429)(317 430)(318 431)(319 432)(320 425)(321 396)(322 397)(323 398)(324 399)(325 400)(326 393)(327 394)(328 395)
(1 232 441 387)(2 225 442 388)(3 226 443 389)(4 227 444 390)(5 228 445 391)(6 229 446 392)(7 230 447 385)(8 231 448 386)(9 185 235 402)(10 186 236 403)(11 187 237 404)(12 188 238 405)(13 189 239 406)(14 190 240 407)(15 191 233 408)(16 192 234 401)(17 248 181 45)(18 241 182 46)(19 242 183 47)(20 243 184 48)(21 244 177 41)(22 245 178 42)(23 246 179 43)(24 247 180 44)(25 256 221 53)(26 249 222 54)(27 250 223 55)(28 251 224 56)(29 252 217 49)(30 253 218 50)(31 254 219 51)(32 255 220 52)(33 263 174 432)(34 264 175 425)(35 257 176 426)(36 258 169 427)(37 259 170 428)(38 260 171 429)(39 261 172 430)(40 262 173 431)(57 86 121 129)(58 87 122 130)(59 88 123 131)(60 81 124 132)(61 82 125 133)(62 83 126 134)(63 84 127 135)(64 85 128 136)(65 360 297 288)(66 353 298 281)(67 354 299 282)(68 355 300 283)(69 356 301 284)(70 357 302 285)(71 358 303 286)(72 359 304 287)(73 116 145 380)(74 117 146 381)(75 118 147 382)(76 119 148 383)(77 120 149 384)(78 113 150 377)(79 114 151 378)(80 115 152 379)(89 168 325 349)(90 161 326 350)(91 162 327 351)(92 163 328 352)(93 164 321 345)(94 165 322 346)(95 166 323 347)(96 167 324 348)(97 331 276 295)(98 332 277 296)(99 333 278 289)(100 334 279 290)(101 335 280 291)(102 336 273 292)(103 329 274 293)(104 330 275 294)(105 371 316 144)(106 372 317 137)(107 373 318 138)(108 374 319 139)(109 375 320 140)(110 376 313 141)(111 369 314 142)(112 370 315 143)(153 340 361 306)(154 341 362 307)(155 342 363 308)(156 343 364 309)(157 344 365 310)(158 337 366 311)(159 338 367 312)(160 339 368 305)(193 270 410 438)(194 271 411 439)(195 272 412 440)(196 265 413 433)(197 266 414 434)(198 267 415 435)(199 268 416 436)(200 269 409 437)(201 214 419 399)(202 215 420 400)(203 216 421 393)(204 209 422 394)(205 210 423 395)(206 211 424 396)(207 212 417 397)(208 213 418 398)
(1 408 411 46 49 202 262)(2 263 203 50 47 412 401)(3 402 413 48 51 204 264)(4 257 205 52 41 414 403)(5 404 415 42 53 206 258)(6 259 207 54 43 416 405)(7 406 409 44 55 208 260)(8 261 201 56 45 410 407)(9 433 20 31 209 175 226)(10 227 176 210 32 21 434)(11 435 22 25 211 169 228)(12 229 170 212 26 23 436)(13 437 24 27 213 171 230)(14 231 172 214 28 17 438)(15 439 18 29 215 173 232)(16 225 174 216 30 19 440)(33 393 218 183 272 234 388)(34 389 235 265 184 219 394)(35 395 220 177 266 236 390)(36 391 237 267 178 221 396)(37 397 222 179 268 238 392)(38 385 239 269 180 223 398)(39 399 224 181 270 240 386)(40 387 233 271 182 217 400)(57 382 273 315 165 339 70)(58 71 340 166 316 274 383)(59 384 275 317 167 341 72)(60 65 342 168 318 276 377)(61 378 277 319 161 343 66)(62 67 344 162 320 278 379)(63 380 279 313 163 337 68)(64 69 338 164 314 280 381)(73 290 141 328 366 355 84)(74 85 356 367 321 142 291)(75 292 143 322 368 357 86)(76 87 358 361 323 144 293)(77 294 137 324 362 359 88)(78 81 360 363 325 138 295)(79 296 139 326 364 353 82)(80 83 354 365 327 140 289)(89 373 331 150 132 288 155)(90 156 281 133 151 332 374)(91 375 333 152 134 282 157)(92 158 283 135 145 334 376)(93 369 335 146 136 284 159)(94 160 285 129 147 336 370)(95 371 329 148 130 286 153)(96 154 287 131 149 330 372)(97 113 124 297 308 349 107)(98 108 350 309 298 125 114)(99 115 126 299 310 351 109)(100 110 352 311 300 127 116)(101 117 128 301 312 345 111)(102 112 346 305 302 121 118)(103 119 122 303 306 347 105)(104 106 348 307 304 123 120)(185 196 243 254 422 425 443)(186 444 426 423 255 244 197)(187 198 245 256 424 427 445)(188 446 428 417 249 246 199)(189 200 247 250 418 429 447)(190 448 430 419 251 248 193)(191 194 241 252 420 431 441)(192 442 432 421 253 242 195)
(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)

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

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

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

160 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 7A 7B 7C 8A ··· 8AF 14A ··· 14U 28A ··· 28BT order 1 2 ··· 2 4 ··· 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 size 1 1 ··· 1 1 ··· 1 2 2 2 7 ··· 7 2 ··· 2 2 ··· 2

160 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + - + - + image C1 C2 C2 C2 C4 C4 C4 C8 D7 Dic7 D14 Dic7 D14 C7⋊C8 C4×D7 kernel C2×C4×C7⋊C8 C4×C7⋊C8 C22×C7⋊C8 C2×C4×C28 C2×C7⋊C8 C4×C28 C22×C28 C2×C28 C2×C42 C42 C42 C22×C4 C22×C4 C2×C4 C2×C4 # reps 1 4 2 1 16 4 4 32 3 6 6 6 3 48 24

Matrix representation of C2×C4×C7⋊C8 in GL4(𝔽113) generated by

 112 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 112 0 0 0 0 98 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 10 112 0 0 11 112
,
 112 0 0 0 0 98 0 0 0 0 97 7 0 0 14 16
G:=sub<GL(4,GF(113))| [112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,98,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,10,11,0,0,112,112],[112,0,0,0,0,98,0,0,0,0,97,14,0,0,7,16] >;

C2×C4×C7⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_7\rtimes C_8
% in TeX

G:=Group("C2xC4xC7:C8");
// GroupNames label

G:=SmallGroup(448,454);
// by ID

G=gap.SmallGroup(448,454);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,100,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^7=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽