direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C28⋊C8, C42.268D14, C42.11Dic7, (C2×C28)⋊5C8, C28⋊7(C2×C8), C14⋊1(C4⋊C8), (C4×C28).16C4, C4.79(C2×D28), C28.44(C4⋊C4), (C2×C28).65Q8, C28.82(C2×Q8), (C2×C42).9D7, C28.299(C2×D4), (C2×C4).166D28, (C2×C28).399D4, C14.22(C22×C8), (C22×C28).22C4, C4.22(C4⋊Dic7), C4.47(C2×Dic14), (C2×C4).55Dic14, (C4×C28).329C22, (C2×C28).841C23, (C22×C4).455D14, (C2×C14).25M4(2), C14.36(C2×M4(2)), C23.40(C2×Dic7), (C22×C4).14Dic7, C22.20(C4⋊Dic7), C22.9(C4.Dic7), (C22×C28).552C22, C22.14(C22×Dic7), C7⋊2(C2×C4⋊C8), C4⋊2(C2×C7⋊C8), (C2×C4)⋊3(C7⋊C8), (C2×C4×C28).17C2, C2.3(C22×C7⋊C8), C14.22(C2×C4⋊C4), C22.13(C2×C7⋊C8), C2.1(C2×C4⋊Dic7), (C2×C14).34(C2×C8), (C22×C7⋊C8).18C2, (C2×C14).42(C4⋊C4), (C2×C28).276(C2×C4), C2.3(C2×C4.Dic7), (C2×C7⋊C8).311C22, (C2×C4).97(C2×Dic7), (C2×C4).783(C22×D7), (C2×C14).170(C22×C4), (C22×C14).129(C2×C4), SmallGroup(448,457)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C28⋊C8
G = < a,b,c | a2=b28=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 324 in 138 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C22×C4, C28, C28, C28, C2×C14, C2×C14, C4⋊C8, C2×C42, C22×C8, C7⋊C8, C2×C28, C2×C28, C2×C28, C22×C14, C2×C4⋊C8, C2×C7⋊C8, C2×C7⋊C8, C4×C28, C22×C28, C28⋊C8, C22×C7⋊C8, C2×C4×C28, C2×C28⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, D7, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, Dic7, D14, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C7⋊C8, Dic14, D28, C2×Dic7, C22×D7, C2×C4⋊C8, C2×C7⋊C8, C4.Dic7, C4⋊Dic7, C2×Dic14, C2×D28, C22×Dic7, C28⋊C8, C22×C7⋊C8, C2×C4.Dic7, C2×C4⋊Dic7, C2×C28⋊C8
►Regular action on 448 points
Generators in S448
(1 326)(2 327)(3 328)(4 329)(5 330)(6 331)(7 332)(8 333)(9 334)(10 335)(11 336)(12 309)(13 310)(14 311)(15 312)(16 313)(17 314)(18 315)(19 316)(20 317)(21 318)(22 319)(23 320)(24 321)(25 322)(26 323)(27 324)(28 325)(29 181)(30 182)(31 183)(32 184)(33 185)(34 186)(35 187)(36 188)(37 189)(38 190)(39 191)(40 192)(41 193)(42 194)(43 195)(44 196)(45 169)(46 170)(47 171)(48 172)(49 173)(50 174)(51 175)(52 176)(53 177)(54 178)(55 179)(56 180)(57 202)(58 203)(59 204)(60 205)(61 206)(62 207)(63 208)(64 209)(65 210)(66 211)(67 212)(68 213)(69 214)(70 215)(71 216)(72 217)(73 218)(74 219)(75 220)(76 221)(77 222)(78 223)(79 224)(80 197)(81 198)(82 199)(83 200)(84 201)(85 347)(86 348)(87 349)(88 350)(89 351)(90 352)(91 353)(92 354)(93 355)(94 356)(95 357)(96 358)(97 359)(98 360)(99 361)(100 362)(101 363)(102 364)(103 337)(104 338)(105 339)(106 340)(107 341)(108 342)(109 343)(110 344)(111 345)(112 346)(113 164)(114 165)(115 166)(116 167)(117 168)(118 141)(119 142)(120 143)(121 144)(122 145)(123 146)(124 147)(125 148)(126 149)(127 150)(128 151)(129 152)(130 153)(131 154)(132 155)(133 156)(134 157)(135 158)(136 159)(137 160)(138 161)(139 162)(140 163)(225 367)(226 368)(227 369)(228 370)(229 371)(230 372)(231 373)(232 374)(233 375)(234 376)(235 377)(236 378)(237 379)(238 380)(239 381)(240 382)(241 383)(242 384)(243 385)(244 386)(245 387)(246 388)(247 389)(248 390)(249 391)(250 392)(251 365)(252 366)(253 399)(254 400)(255 401)(256 402)(257 403)(258 404)(259 405)(260 406)(261 407)(262 408)(263 409)(264 410)(265 411)(266 412)(267 413)(268 414)(269 415)(270 416)(271 417)(272 418)(273 419)(274 420)(275 393)(276 394)(277 395)(278 396)(279 397)(280 398)(281 448)(282 421)(283 422)(284 423)(285 424)(286 425)(287 426)(288 427)(289 428)(290 429)(291 430)(292 431)(293 432)(294 433)(295 434)(296 435)(297 436)(298 437)(299 438)(300 439)(301 440)(302 441)(303 442)(304 443)(305 444)(306 445)(307 446)(308 447)
(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)
(1 403 441 211 382 164 31 97)(2 402 442 210 383 163 32 96)(3 401 443 209 384 162 33 95)(4 400 444 208 385 161 34 94)(5 399 445 207 386 160 35 93)(6 398 446 206 387 159 36 92)(7 397 447 205 388 158 37 91)(8 396 448 204 389 157 38 90)(9 395 421 203 390 156 39 89)(10 394 422 202 391 155 40 88)(11 393 423 201 392 154 41 87)(12 420 424 200 365 153 42 86)(13 419 425 199 366 152 43 85)(14 418 426 198 367 151 44 112)(15 417 427 197 368 150 45 111)(16 416 428 224 369 149 46 110)(17 415 429 223 370 148 47 109)(18 414 430 222 371 147 48 108)(19 413 431 221 372 146 49 107)(20 412 432 220 373 145 50 106)(21 411 433 219 374 144 51 105)(22 410 434 218 375 143 52 104)(23 409 435 217 376 142 53 103)(24 408 436 216 377 141 54 102)(25 407 437 215 378 168 55 101)(26 406 438 214 379 167 56 100)(27 405 439 213 380 166 29 99)(28 404 440 212 381 165 30 98)(57 249 132 192 350 335 276 283)(58 248 133 191 351 334 277 282)(59 247 134 190 352 333 278 281)(60 246 135 189 353 332 279 308)(61 245 136 188 354 331 280 307)(62 244 137 187 355 330 253 306)(63 243 138 186 356 329 254 305)(64 242 139 185 357 328 255 304)(65 241 140 184 358 327 256 303)(66 240 113 183 359 326 257 302)(67 239 114 182 360 325 258 301)(68 238 115 181 361 324 259 300)(69 237 116 180 362 323 260 299)(70 236 117 179 363 322 261 298)(71 235 118 178 364 321 262 297)(72 234 119 177 337 320 263 296)(73 233 120 176 338 319 264 295)(74 232 121 175 339 318 265 294)(75 231 122 174 340 317 266 293)(76 230 123 173 341 316 267 292)(77 229 124 172 342 315 268 291)(78 228 125 171 343 314 269 290)(79 227 126 170 344 313 270 289)(80 226 127 169 345 312 271 288)(81 225 128 196 346 311 272 287)(82 252 129 195 347 310 273 286)(83 251 130 194 348 309 274 285)(84 250 131 193 349 336 275 284)
G:=sub<Sym(448)| (1,326)(2,327)(3,328)(4,329)(5,330)(6,331)(7,332)(8,333)(9,334)(10,335)(11,336)(12,309)(13,310)(14,311)(15,312)(16,313)(17,314)(18,315)(19,316)(20,317)(21,318)(22,319)(23,320)(24,321)(25,322)(26,323)(27,324)(28,325)(29,181)(30,182)(31,183)(32,184)(33,185)(34,186)(35,187)(36,188)(37,189)(38,190)(39,191)(40,192)(41,193)(42,194)(43,195)(44,196)(45,169)(46,170)(47,171)(48,172)(49,173)(50,174)(51,175)(52,176)(53,177)(54,178)(55,179)(56,180)(57,202)(58,203)(59,204)(60,205)(61,206)(62,207)(63,208)(64,209)(65,210)(66,211)(67,212)(68,213)(69,214)(70,215)(71,216)(72,217)(73,218)(74,219)(75,220)(76,221)(77,222)(78,223)(79,224)(80,197)(81,198)(82,199)(83,200)(84,201)(85,347)(86,348)(87,349)(88,350)(89,351)(90,352)(91,353)(92,354)(93,355)(94,356)(95,357)(96,358)(97,359)(98,360)(99,361)(100,362)(101,363)(102,364)(103,337)(104,338)(105,339)(106,340)(107,341)(108,342)(109,343)(110,344)(111,345)(112,346)(113,164)(114,165)(115,166)(116,167)(117,168)(118,141)(119,142)(120,143)(121,144)(122,145)(123,146)(124,147)(125,148)(126,149)(127,150)(128,151)(129,152)(130,153)(131,154)(132,155)(133,156)(134,157)(135,158)(136,159)(137,160)(138,161)(139,162)(140,163)(225,367)(226,368)(227,369)(228,370)(229,371)(230,372)(231,373)(232,374)(233,375)(234,376)(235,377)(236,378)(237,379)(238,380)(239,381)(240,382)(241,383)(242,384)(243,385)(244,386)(245,387)(246,388)(247,389)(248,390)(249,391)(250,392)(251,365)(252,366)(253,399)(254,400)(255,401)(256,402)(257,403)(258,404)(259,405)(260,406)(261,407)(262,408)(263,409)(264,410)(265,411)(266,412)(267,413)(268,414)(269,415)(270,416)(271,417)(272,418)(273,419)(274,420)(275,393)(276,394)(277,395)(278,396)(279,397)(280,398)(281,448)(282,421)(283,422)(284,423)(285,424)(286,425)(287,426)(288,427)(289,428)(290,429)(291,430)(292,431)(293,432)(294,433)(295,434)(296,435)(297,436)(298,437)(299,438)(300,439)(301,440)(302,441)(303,442)(304,443)(305,444)(306,445)(307,446)(308,447), (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), (1,403,441,211,382,164,31,97)(2,402,442,210,383,163,32,96)(3,401,443,209,384,162,33,95)(4,400,444,208,385,161,34,94)(5,399,445,207,386,160,35,93)(6,398,446,206,387,159,36,92)(7,397,447,205,388,158,37,91)(8,396,448,204,389,157,38,90)(9,395,421,203,390,156,39,89)(10,394,422,202,391,155,40,88)(11,393,423,201,392,154,41,87)(12,420,424,200,365,153,42,86)(13,419,425,199,366,152,43,85)(14,418,426,198,367,151,44,112)(15,417,427,197,368,150,45,111)(16,416,428,224,369,149,46,110)(17,415,429,223,370,148,47,109)(18,414,430,222,371,147,48,108)(19,413,431,221,372,146,49,107)(20,412,432,220,373,145,50,106)(21,411,433,219,374,144,51,105)(22,410,434,218,375,143,52,104)(23,409,435,217,376,142,53,103)(24,408,436,216,377,141,54,102)(25,407,437,215,378,168,55,101)(26,406,438,214,379,167,56,100)(27,405,439,213,380,166,29,99)(28,404,440,212,381,165,30,98)(57,249,132,192,350,335,276,283)(58,248,133,191,351,334,277,282)(59,247,134,190,352,333,278,281)(60,246,135,189,353,332,279,308)(61,245,136,188,354,331,280,307)(62,244,137,187,355,330,253,306)(63,243,138,186,356,329,254,305)(64,242,139,185,357,328,255,304)(65,241,140,184,358,327,256,303)(66,240,113,183,359,326,257,302)(67,239,114,182,360,325,258,301)(68,238,115,181,361,324,259,300)(69,237,116,180,362,323,260,299)(70,236,117,179,363,322,261,298)(71,235,118,178,364,321,262,297)(72,234,119,177,337,320,263,296)(73,233,120,176,338,319,264,295)(74,232,121,175,339,318,265,294)(75,231,122,174,340,317,266,293)(76,230,123,173,341,316,267,292)(77,229,124,172,342,315,268,291)(78,228,125,171,343,314,269,290)(79,227,126,170,344,313,270,289)(80,226,127,169,345,312,271,288)(81,225,128,196,346,311,272,287)(82,252,129,195,347,310,273,286)(83,251,130,194,348,309,274,285)(84,250,131,193,349,336,275,284)>;
G:=Group( (1,326)(2,327)(3,328)(4,329)(5,330)(6,331)(7,332)(8,333)(9,334)(10,335)(11,336)(12,309)(13,310)(14,311)(15,312)(16,313)(17,314)(18,315)(19,316)(20,317)(21,318)(22,319)(23,320)(24,321)(25,322)(26,323)(27,324)(28,325)(29,181)(30,182)(31,183)(32,184)(33,185)(34,186)(35,187)(36,188)(37,189)(38,190)(39,191)(40,192)(41,193)(42,194)(43,195)(44,196)(45,169)(46,170)(47,171)(48,172)(49,173)(50,174)(51,175)(52,176)(53,177)(54,178)(55,179)(56,180)(57,202)(58,203)(59,204)(60,205)(61,206)(62,207)(63,208)(64,209)(65,210)(66,211)(67,212)(68,213)(69,214)(70,215)(71,216)(72,217)(73,218)(74,219)(75,220)(76,221)(77,222)(78,223)(79,224)(80,197)(81,198)(82,199)(83,200)(84,201)(85,347)(86,348)(87,349)(88,350)(89,351)(90,352)(91,353)(92,354)(93,355)(94,356)(95,357)(96,358)(97,359)(98,360)(99,361)(100,362)(101,363)(102,364)(103,337)(104,338)(105,339)(106,340)(107,341)(108,342)(109,343)(110,344)(111,345)(112,346)(113,164)(114,165)(115,166)(116,167)(117,168)(118,141)(119,142)(120,143)(121,144)(122,145)(123,146)(124,147)(125,148)(126,149)(127,150)(128,151)(129,152)(130,153)(131,154)(132,155)(133,156)(134,157)(135,158)(136,159)(137,160)(138,161)(139,162)(140,163)(225,367)(226,368)(227,369)(228,370)(229,371)(230,372)(231,373)(232,374)(233,375)(234,376)(235,377)(236,378)(237,379)(238,380)(239,381)(240,382)(241,383)(242,384)(243,385)(244,386)(245,387)(246,388)(247,389)(248,390)(249,391)(250,392)(251,365)(252,366)(253,399)(254,400)(255,401)(256,402)(257,403)(258,404)(259,405)(260,406)(261,407)(262,408)(263,409)(264,410)(265,411)(266,412)(267,413)(268,414)(269,415)(270,416)(271,417)(272,418)(273,419)(274,420)(275,393)(276,394)(277,395)(278,396)(279,397)(280,398)(281,448)(282,421)(283,422)(284,423)(285,424)(286,425)(287,426)(288,427)(289,428)(290,429)(291,430)(292,431)(293,432)(294,433)(295,434)(296,435)(297,436)(298,437)(299,438)(300,439)(301,440)(302,441)(303,442)(304,443)(305,444)(306,445)(307,446)(308,447), (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), (1,403,441,211,382,164,31,97)(2,402,442,210,383,163,32,96)(3,401,443,209,384,162,33,95)(4,400,444,208,385,161,34,94)(5,399,445,207,386,160,35,93)(6,398,446,206,387,159,36,92)(7,397,447,205,388,158,37,91)(8,396,448,204,389,157,38,90)(9,395,421,203,390,156,39,89)(10,394,422,202,391,155,40,88)(11,393,423,201,392,154,41,87)(12,420,424,200,365,153,42,86)(13,419,425,199,366,152,43,85)(14,418,426,198,367,151,44,112)(15,417,427,197,368,150,45,111)(16,416,428,224,369,149,46,110)(17,415,429,223,370,148,47,109)(18,414,430,222,371,147,48,108)(19,413,431,221,372,146,49,107)(20,412,432,220,373,145,50,106)(21,411,433,219,374,144,51,105)(22,410,434,218,375,143,52,104)(23,409,435,217,376,142,53,103)(24,408,436,216,377,141,54,102)(25,407,437,215,378,168,55,101)(26,406,438,214,379,167,56,100)(27,405,439,213,380,166,29,99)(28,404,440,212,381,165,30,98)(57,249,132,192,350,335,276,283)(58,248,133,191,351,334,277,282)(59,247,134,190,352,333,278,281)(60,246,135,189,353,332,279,308)(61,245,136,188,354,331,280,307)(62,244,137,187,355,330,253,306)(63,243,138,186,356,329,254,305)(64,242,139,185,357,328,255,304)(65,241,140,184,358,327,256,303)(66,240,113,183,359,326,257,302)(67,239,114,182,360,325,258,301)(68,238,115,181,361,324,259,300)(69,237,116,180,362,323,260,299)(70,236,117,179,363,322,261,298)(71,235,118,178,364,321,262,297)(72,234,119,177,337,320,263,296)(73,233,120,176,338,319,264,295)(74,232,121,175,339,318,265,294)(75,231,122,174,340,317,266,293)(76,230,123,173,341,316,267,292)(77,229,124,172,342,315,268,291)(78,228,125,171,343,314,269,290)(79,227,126,170,344,313,270,289)(80,226,127,169,345,312,271,288)(81,225,128,196,346,311,272,287)(82,252,129,195,347,310,273,286)(83,251,130,194,348,309,274,285)(84,250,131,193,349,336,275,284) );
G=PermutationGroup([[(1,326),(2,327),(3,328),(4,329),(5,330),(6,331),(7,332),(8,333),(9,334),(10,335),(11,336),(12,309),(13,310),(14,311),(15,312),(16,313),(17,314),(18,315),(19,316),(20,317),(21,318),(22,319),(23,320),(24,321),(25,322),(26,323),(27,324),(28,325),(29,181),(30,182),(31,183),(32,184),(33,185),(34,186),(35,187),(36,188),(37,189),(38,190),(39,191),(40,192),(41,193),(42,194),(43,195),(44,196),(45,169),(46,170),(47,171),(48,172),(49,173),(50,174),(51,175),(52,176),(53,177),(54,178),(55,179),(56,180),(57,202),(58,203),(59,204),(60,205),(61,206),(62,207),(63,208),(64,209),(65,210),(66,211),(67,212),(68,213),(69,214),(70,215),(71,216),(72,217),(73,218),(74,219),(75,220),(76,221),(77,222),(78,223),(79,224),(80,197),(81,198),(82,199),(83,200),(84,201),(85,347),(86,348),(87,349),(88,350),(89,351),(90,352),(91,353),(92,354),(93,355),(94,356),(95,357),(96,358),(97,359),(98,360),(99,361),(100,362),(101,363),(102,364),(103,337),(104,338),(105,339),(106,340),(107,341),(108,342),(109,343),(110,344),(111,345),(112,346),(113,164),(114,165),(115,166),(116,167),(117,168),(118,141),(119,142),(120,143),(121,144),(122,145),(123,146),(124,147),(125,148),(126,149),(127,150),(128,151),(129,152),(130,153),(131,154),(132,155),(133,156),(134,157),(135,158),(136,159),(137,160),(138,161),(139,162),(140,163),(225,367),(226,368),(227,369),(228,370),(229,371),(230,372),(231,373),(232,374),(233,375),(234,376),(235,377),(236,378),(237,379),(238,380),(239,381),(240,382),(241,383),(242,384),(243,385),(244,386),(245,387),(246,388),(247,389),(248,390),(249,391),(250,392),(251,365),(252,366),(253,399),(254,400),(255,401),(256,402),(257,403),(258,404),(259,405),(260,406),(261,407),(262,408),(263,409),(264,410),(265,411),(266,412),(267,413),(268,414),(269,415),(270,416),(271,417),(272,418),(273,419),(274,420),(275,393),(276,394),(277,395),(278,396),(279,397),(280,398),(281,448),(282,421),(283,422),(284,423),(285,424),(286,425),(287,426),(288,427),(289,428),(290,429),(291,430),(292,431),(293,432),(294,433),(295,434),(296,435),(297,436),(298,437),(299,438),(300,439),(301,440),(302,441),(303,442),(304,443),(305,444),(306,445),(307,446),(308,447)], [(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)], [(1,403,441,211,382,164,31,97),(2,402,442,210,383,163,32,96),(3,401,443,209,384,162,33,95),(4,400,444,208,385,161,34,94),(5,399,445,207,386,160,35,93),(6,398,446,206,387,159,36,92),(7,397,447,205,388,158,37,91),(8,396,448,204,389,157,38,90),(9,395,421,203,390,156,39,89),(10,394,422,202,391,155,40,88),(11,393,423,201,392,154,41,87),(12,420,424,200,365,153,42,86),(13,419,425,199,366,152,43,85),(14,418,426,198,367,151,44,112),(15,417,427,197,368,150,45,111),(16,416,428,224,369,149,46,110),(17,415,429,223,370,148,47,109),(18,414,430,222,371,147,48,108),(19,413,431,221,372,146,49,107),(20,412,432,220,373,145,50,106),(21,411,433,219,374,144,51,105),(22,410,434,218,375,143,52,104),(23,409,435,217,376,142,53,103),(24,408,436,216,377,141,54,102),(25,407,437,215,378,168,55,101),(26,406,438,214,379,167,56,100),(27,405,439,213,380,166,29,99),(28,404,440,212,381,165,30,98),(57,249,132,192,350,335,276,283),(58,248,133,191,351,334,277,282),(59,247,134,190,352,333,278,281),(60,246,135,189,353,332,279,308),(61,245,136,188,354,331,280,307),(62,244,137,187,355,330,253,306),(63,243,138,186,356,329,254,305),(64,242,139,185,357,328,255,304),(65,241,140,184,358,327,256,303),(66,240,113,183,359,326,257,302),(67,239,114,182,360,325,258,301),(68,238,115,181,361,324,259,300),(69,237,116,180,362,323,260,299),(70,236,117,179,363,322,261,298),(71,235,118,178,364,321,262,297),(72,234,119,177,337,320,263,296),(73,233,120,176,338,319,264,295),(74,232,121,175,339,318,265,294),(75,231,122,174,340,317,266,293),(76,230,123,173,341,316,267,292),(77,229,124,172,342,315,268,291),(78,228,125,171,343,314,269,290),(79,227,126,170,344,313,270,289),(80,226,127,169,345,312,271,288),(81,225,128,196,346,311,272,287),(82,252,129,195,347,310,273,286),(83,251,130,194,348,309,274,285),(84,250,131,193,349,336,275,284)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 8A | ··· | 8P | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D7 | M4(2) | Dic7 | D14 | Dic7 | D14 | C7⋊C8 | Dic14 | D28 | C4.Dic7 |
| kernel | C2×C28⋊C8 | C28⋊C8 | C22×C7⋊C8 | C2×C4×C28 | C4×C28 | C22×C28 | C2×C28 | C2×C28 | C2×C28 | C2×C42 | C2×C14 | C42 | C42 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 4 | 4 | 16 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 3 | 24 | 12 | 12 | 24 |
Matrix representation of C2×C28⋊C8 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 79 | 24 |
| 0 | 0 | 0 | 54 | 25 |
| 15 | 0 | 0 | 0 | 0 |
| 0 | 61 | 81 | 0 | 0 |
| 0 | 81 | 52 | 0 | 0 |
| 0 | 0 | 0 | 48 | 82 |
| 0 | 0 | 0 | 103 | 65 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,1,0,0,0,0,0,0,79,54,0,0,0,24,25],[15,0,0,0,0,0,61,81,0,0,0,81,52,0,0,0,0,0,48,103,0,0,0,82,65] >;
C2×C28⋊C8 in GAP, Magma, Sage, TeX
C_2\times C_{28}\rtimes C_8 % in TeX
G:=Group("C2xC28:C8"); // GroupNames label
G:=SmallGroup(448,457);
// by ID
G=gap.SmallGroup(448,457);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,100,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^2=b^28=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations