direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8⋊Dic7, C23.56D28, (C2×C56)⋊11C4, C56⋊30(C2×C4), C8⋊8(C2×Dic7), (C2×C8)⋊7Dic7, (C2×C4).93D28, C14⋊2(C4.Q8), C28.35(C4⋊C4), C28.73(C2×Q8), (C2×C28).55Q8, (C2×C28).386D4, (C2×C8).319D14, (C22×C8).13D7, (C22×C56).19C2, C4.16(C4⋊Dic7), C14.15(C2×SD16), (C2×C4).48Dic14, (C2×C14).21SD16, C4.39(C2×Dic14), C22.50(C2×D28), (C2×C56).391C22, (C2×C28).763C23, C28.169(C22×C4), (C22×C4).423D14, (C22×C14).135D4, C4.23(C22×Dic7), C22.11(C56⋊C2), C4⋊Dic7.279C22, C22.21(C4⋊Dic7), (C22×C28).515C22, C7⋊3(C2×C4.Q8), C14.44(C2×C4⋊C4), C2.3(C2×C56⋊C2), C2.10(C2×C4⋊Dic7), (C2×C14).49(C4⋊C4), (C2×C28).298(C2×C4), (C2×C14).153(C2×D4), (C2×C4⋊Dic7).22C2, (C2×C4).81(C2×Dic7), (C2×C4).710(C22×D7), SmallGroup(448,638)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C8⋊Dic7
G = < a,b,c,d | a2=b8=c14=1, d2=c7, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c-1 >
Subgroups: 548 in 130 conjugacy classes, 87 normal (21 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C4.Q8, C2×C4⋊C4, C22×C8, C56, C2×Dic7, C2×C28, C2×C28, C22×C14, C2×C4.Q8, C4⋊Dic7, C4⋊Dic7, C2×C56, C22×Dic7, C22×C28, C8⋊Dic7, C2×C4⋊Dic7, C22×C56, C2×C8⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, Dic7, D14, C4.Q8, C2×C4⋊C4, C2×SD16, Dic14, D28, C2×Dic7, C22×D7, C2×C4.Q8, C56⋊C2, C4⋊Dic7, C2×Dic14, C2×D28, C22×Dic7, C8⋊Dic7, C2×C56⋊C2, C2×C4⋊Dic7, C2×C8⋊Dic7
►Regular action on 448 points
Generators in S448
(1 54)(2 55)(3 56)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 384)(16 385)(17 386)(18 387)(19 388)(20 389)(21 390)(22 391)(23 392)(24 379)(25 380)(26 381)(27 382)(28 383)(29 140)(30 127)(31 128)(32 129)(33 130)(34 131)(35 132)(36 133)(37 134)(38 135)(39 136)(40 137)(41 138)(42 139)(57 368)(58 369)(59 370)(60 371)(61 372)(62 373)(63 374)(64 375)(65 376)(66 377)(67 378)(68 365)(69 366)(70 367)(71 267)(72 268)(73 269)(74 270)(75 271)(76 272)(77 273)(78 274)(79 275)(80 276)(81 277)(82 278)(83 279)(84 280)(85 363)(86 364)(87 351)(88 352)(89 353)(90 354)(91 355)(92 356)(93 357)(94 358)(95 359)(96 360)(97 361)(98 362)(99 184)(100 185)(101 186)(102 187)(103 188)(104 189)(105 190)(106 191)(107 192)(108 193)(109 194)(110 195)(111 196)(112 183)(113 421)(114 422)(115 423)(116 424)(117 425)(118 426)(119 427)(120 428)(121 429)(122 430)(123 431)(124 432)(125 433)(126 434)(141 208)(142 209)(143 210)(144 197)(145 198)(146 199)(147 200)(148 201)(149 202)(150 203)(151 204)(152 205)(153 206)(154 207)(155 256)(156 257)(157 258)(158 259)(159 260)(160 261)(161 262)(162 263)(163 264)(164 265)(165 266)(166 253)(167 254)(168 255)(169 288)(170 289)(171 290)(172 291)(173 292)(174 293)(175 294)(176 281)(177 282)(178 283)(179 284)(180 285)(181 286)(182 287)(211 323)(212 324)(213 325)(214 326)(215 327)(216 328)(217 329)(218 330)(219 331)(220 332)(221 333)(222 334)(223 335)(224 336)(225 411)(226 412)(227 413)(228 414)(229 415)(230 416)(231 417)(232 418)(233 419)(234 420)(235 407)(236 408)(237 409)(238 410)(239 439)(240 440)(241 441)(242 442)(243 443)(244 444)(245 445)(246 446)(247 447)(248 448)(249 435)(250 436)(251 437)(252 438)(295 317)(296 318)(297 319)(298 320)(299 321)(300 322)(301 309)(302 310)(303 311)(304 312)(305 313)(306 314)(307 315)(308 316)(337 397)(338 398)(339 399)(340 400)(341 401)(342 402)(343 403)(344 404)(345 405)(346 406)(347 393)(348 394)(349 395)(350 396)
(1 15 417 64 42 267 92 158)(2 16 418 65 29 268 93 159)(3 17 419 66 30 269 94 160)(4 18 420 67 31 270 95 161)(5 19 407 68 32 271 96 162)(6 20 408 69 33 272 97 163)(7 21 409 70 34 273 98 164)(8 22 410 57 35 274 85 165)(9 23 411 58 36 275 86 166)(10 24 412 59 37 276 87 167)(11 25 413 60 38 277 88 168)(12 26 414 61 39 278 89 155)(13 27 415 62 40 279 90 156)(14 28 416 63 41 280 91 157)(43 387 234 378 128 74 359 262)(44 388 235 365 129 75 360 263)(45 389 236 366 130 76 361 264)(46 390 237 367 131 77 362 265)(47 391 238 368 132 78 363 266)(48 392 225 369 133 79 364 253)(49 379 226 370 134 80 351 254)(50 380 227 371 135 81 352 255)(51 381 228 372 136 82 353 256)(52 382 229 373 137 83 354 257)(53 383 230 374 138 84 355 258)(54 384 231 375 139 71 356 259)(55 385 232 376 140 72 357 260)(56 386 233 377 127 73 358 261)(99 445 426 282 210 343 310 335)(100 446 427 283 197 344 311 336)(101 447 428 284 198 345 312 323)(102 448 429 285 199 346 313 324)(103 435 430 286 200 347 314 325)(104 436 431 287 201 348 315 326)(105 437 432 288 202 349 316 327)(106 438 433 289 203 350 317 328)(107 439 434 290 204 337 318 329)(108 440 421 291 205 338 319 330)(109 441 422 292 206 339 320 331)(110 442 423 293 207 340 321 332)(111 443 424 294 208 341 322 333)(112 444 425 281 209 342 309 334)(113 172 152 398 297 218 193 240)(114 173 153 399 298 219 194 241)(115 174 154 400 299 220 195 242)(116 175 141 401 300 221 196 243)(117 176 142 402 301 222 183 244)(118 177 143 403 302 223 184 245)(119 178 144 404 303 224 185 246)(120 179 145 405 304 211 186 247)(121 180 146 406 305 212 187 248)(122 181 147 393 306 213 188 249)(123 182 148 394 307 214 189 250)(124 169 149 395 308 215 190 251)(125 170 150 396 295 216 191 252)(126 171 151 397 296 217 192 239)
(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)
(1 428 8 421)(2 427 9 434)(3 426 10 433)(4 425 11 432)(5 424 12 431)(6 423 13 430)(7 422 14 429)(15 345 22 338)(16 344 23 337)(17 343 24 350)(18 342 25 349)(19 341 26 348)(20 340 27 347)(21 339 28 346)(29 311 36 318)(30 310 37 317)(31 309 38 316)(32 322 39 315)(33 321 40 314)(34 320 41 313)(35 319 42 312)(43 117 50 124)(44 116 51 123)(45 115 52 122)(46 114 53 121)(47 113 54 120)(48 126 55 119)(49 125 56 118)(57 291 64 284)(58 290 65 283)(59 289 66 282)(60 288 67 281)(61 287 68 294)(62 286 69 293)(63 285 70 292)(71 247 78 240)(72 246 79 239)(73 245 80 252)(74 244 81 251)(75 243 82 250)(76 242 83 249)(77 241 84 248)(85 205 92 198)(86 204 93 197)(87 203 94 210)(88 202 95 209)(89 201 96 208)(90 200 97 207)(91 199 98 206)(99 412 106 419)(100 411 107 418)(101 410 108 417)(102 409 109 416)(103 408 110 415)(104 407 111 414)(105 420 112 413)(127 302 134 295)(128 301 135 308)(129 300 136 307)(130 299 137 306)(131 298 138 305)(132 297 139 304)(133 296 140 303)(141 353 148 360)(142 352 149 359)(143 351 150 358)(144 364 151 357)(145 363 152 356)(146 362 153 355)(147 361 154 354)(155 326 162 333)(156 325 163 332)(157 324 164 331)(158 323 165 330)(159 336 166 329)(160 335 167 328)(161 334 168 327)(169 378 176 371)(170 377 177 370)(171 376 178 369)(172 375 179 368)(173 374 180 367)(174 373 181 366)(175 372 182 365)(183 227 190 234)(184 226 191 233)(185 225 192 232)(186 238 193 231)(187 237 194 230)(188 236 195 229)(189 235 196 228)(211 266 218 259)(212 265 219 258)(213 264 220 257)(214 263 221 256)(215 262 222 255)(216 261 223 254)(217 260 224 253)(267 447 274 440)(268 446 275 439)(269 445 276 438)(270 444 277 437)(271 443 278 436)(272 442 279 435)(273 441 280 448)(379 396 386 403)(380 395 387 402)(381 394 388 401)(382 393 389 400)(383 406 390 399)(384 405 391 398)(385 404 392 397)
G:=sub<Sym(448)| (1,54)(2,55)(3,56)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,384)(16,385)(17,386)(18,387)(19,388)(20,389)(21,390)(22,391)(23,392)(24,379)(25,380)(26,381)(27,382)(28,383)(29,140)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(57,368)(58,369)(59,370)(60,371)(61,372)(62,373)(63,374)(64,375)(65,376)(66,377)(67,378)(68,365)(69,366)(70,367)(71,267)(72,268)(73,269)(74,270)(75,271)(76,272)(77,273)(78,274)(79,275)(80,276)(81,277)(82,278)(83,279)(84,280)(85,363)(86,364)(87,351)(88,352)(89,353)(90,354)(91,355)(92,356)(93,357)(94,358)(95,359)(96,360)(97,361)(98,362)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,190)(106,191)(107,192)(108,193)(109,194)(110,195)(111,196)(112,183)(113,421)(114,422)(115,423)(116,424)(117,425)(118,426)(119,427)(120,428)(121,429)(122,430)(123,431)(124,432)(125,433)(126,434)(141,208)(142,209)(143,210)(144,197)(145,198)(146,199)(147,200)(148,201)(149,202)(150,203)(151,204)(152,205)(153,206)(154,207)(155,256)(156,257)(157,258)(158,259)(159,260)(160,261)(161,262)(162,263)(163,264)(164,265)(165,266)(166,253)(167,254)(168,255)(169,288)(170,289)(171,290)(172,291)(173,292)(174,293)(175,294)(176,281)(177,282)(178,283)(179,284)(180,285)(181,286)(182,287)(211,323)(212,324)(213,325)(214,326)(215,327)(216,328)(217,329)(218,330)(219,331)(220,332)(221,333)(222,334)(223,335)(224,336)(225,411)(226,412)(227,413)(228,414)(229,415)(230,416)(231,417)(232,418)(233,419)(234,420)(235,407)(236,408)(237,409)(238,410)(239,439)(240,440)(241,441)(242,442)(243,443)(244,444)(245,445)(246,446)(247,447)(248,448)(249,435)(250,436)(251,437)(252,438)(295,317)(296,318)(297,319)(298,320)(299,321)(300,322)(301,309)(302,310)(303,311)(304,312)(305,313)(306,314)(307,315)(308,316)(337,397)(338,398)(339,399)(340,400)(341,401)(342,402)(343,403)(344,404)(345,405)(346,406)(347,393)(348,394)(349,395)(350,396), (1,15,417,64,42,267,92,158)(2,16,418,65,29,268,93,159)(3,17,419,66,30,269,94,160)(4,18,420,67,31,270,95,161)(5,19,407,68,32,271,96,162)(6,20,408,69,33,272,97,163)(7,21,409,70,34,273,98,164)(8,22,410,57,35,274,85,165)(9,23,411,58,36,275,86,166)(10,24,412,59,37,276,87,167)(11,25,413,60,38,277,88,168)(12,26,414,61,39,278,89,155)(13,27,415,62,40,279,90,156)(14,28,416,63,41,280,91,157)(43,387,234,378,128,74,359,262)(44,388,235,365,129,75,360,263)(45,389,236,366,130,76,361,264)(46,390,237,367,131,77,362,265)(47,391,238,368,132,78,363,266)(48,392,225,369,133,79,364,253)(49,379,226,370,134,80,351,254)(50,380,227,371,135,81,352,255)(51,381,228,372,136,82,353,256)(52,382,229,373,137,83,354,257)(53,383,230,374,138,84,355,258)(54,384,231,375,139,71,356,259)(55,385,232,376,140,72,357,260)(56,386,233,377,127,73,358,261)(99,445,426,282,210,343,310,335)(100,446,427,283,197,344,311,336)(101,447,428,284,198,345,312,323)(102,448,429,285,199,346,313,324)(103,435,430,286,200,347,314,325)(104,436,431,287,201,348,315,326)(105,437,432,288,202,349,316,327)(106,438,433,289,203,350,317,328)(107,439,434,290,204,337,318,329)(108,440,421,291,205,338,319,330)(109,441,422,292,206,339,320,331)(110,442,423,293,207,340,321,332)(111,443,424,294,208,341,322,333)(112,444,425,281,209,342,309,334)(113,172,152,398,297,218,193,240)(114,173,153,399,298,219,194,241)(115,174,154,400,299,220,195,242)(116,175,141,401,300,221,196,243)(117,176,142,402,301,222,183,244)(118,177,143,403,302,223,184,245)(119,178,144,404,303,224,185,246)(120,179,145,405,304,211,186,247)(121,180,146,406,305,212,187,248)(122,181,147,393,306,213,188,249)(123,182,148,394,307,214,189,250)(124,169,149,395,308,215,190,251)(125,170,150,396,295,216,191,252)(126,171,151,397,296,217,192,239), (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), (1,428,8,421)(2,427,9,434)(3,426,10,433)(4,425,11,432)(5,424,12,431)(6,423,13,430)(7,422,14,429)(15,345,22,338)(16,344,23,337)(17,343,24,350)(18,342,25,349)(19,341,26,348)(20,340,27,347)(21,339,28,346)(29,311,36,318)(30,310,37,317)(31,309,38,316)(32,322,39,315)(33,321,40,314)(34,320,41,313)(35,319,42,312)(43,117,50,124)(44,116,51,123)(45,115,52,122)(46,114,53,121)(47,113,54,120)(48,126,55,119)(49,125,56,118)(57,291,64,284)(58,290,65,283)(59,289,66,282)(60,288,67,281)(61,287,68,294)(62,286,69,293)(63,285,70,292)(71,247,78,240)(72,246,79,239)(73,245,80,252)(74,244,81,251)(75,243,82,250)(76,242,83,249)(77,241,84,248)(85,205,92,198)(86,204,93,197)(87,203,94,210)(88,202,95,209)(89,201,96,208)(90,200,97,207)(91,199,98,206)(99,412,106,419)(100,411,107,418)(101,410,108,417)(102,409,109,416)(103,408,110,415)(104,407,111,414)(105,420,112,413)(127,302,134,295)(128,301,135,308)(129,300,136,307)(130,299,137,306)(131,298,138,305)(132,297,139,304)(133,296,140,303)(141,353,148,360)(142,352,149,359)(143,351,150,358)(144,364,151,357)(145,363,152,356)(146,362,153,355)(147,361,154,354)(155,326,162,333)(156,325,163,332)(157,324,164,331)(158,323,165,330)(159,336,166,329)(160,335,167,328)(161,334,168,327)(169,378,176,371)(170,377,177,370)(171,376,178,369)(172,375,179,368)(173,374,180,367)(174,373,181,366)(175,372,182,365)(183,227,190,234)(184,226,191,233)(185,225,192,232)(186,238,193,231)(187,237,194,230)(188,236,195,229)(189,235,196,228)(211,266,218,259)(212,265,219,258)(213,264,220,257)(214,263,221,256)(215,262,222,255)(216,261,223,254)(217,260,224,253)(267,447,274,440)(268,446,275,439)(269,445,276,438)(270,444,277,437)(271,443,278,436)(272,442,279,435)(273,441,280,448)(379,396,386,403)(380,395,387,402)(381,394,388,401)(382,393,389,400)(383,406,390,399)(384,405,391,398)(385,404,392,397)>;
G:=Group( (1,54)(2,55)(3,56)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,384)(16,385)(17,386)(18,387)(19,388)(20,389)(21,390)(22,391)(23,392)(24,379)(25,380)(26,381)(27,382)(28,383)(29,140)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(57,368)(58,369)(59,370)(60,371)(61,372)(62,373)(63,374)(64,375)(65,376)(66,377)(67,378)(68,365)(69,366)(70,367)(71,267)(72,268)(73,269)(74,270)(75,271)(76,272)(77,273)(78,274)(79,275)(80,276)(81,277)(82,278)(83,279)(84,280)(85,363)(86,364)(87,351)(88,352)(89,353)(90,354)(91,355)(92,356)(93,357)(94,358)(95,359)(96,360)(97,361)(98,362)(99,184)(100,185)(101,186)(102,187)(103,188)(104,189)(105,190)(106,191)(107,192)(108,193)(109,194)(110,195)(111,196)(112,183)(113,421)(114,422)(115,423)(116,424)(117,425)(118,426)(119,427)(120,428)(121,429)(122,430)(123,431)(124,432)(125,433)(126,434)(141,208)(142,209)(143,210)(144,197)(145,198)(146,199)(147,200)(148,201)(149,202)(150,203)(151,204)(152,205)(153,206)(154,207)(155,256)(156,257)(157,258)(158,259)(159,260)(160,261)(161,262)(162,263)(163,264)(164,265)(165,266)(166,253)(167,254)(168,255)(169,288)(170,289)(171,290)(172,291)(173,292)(174,293)(175,294)(176,281)(177,282)(178,283)(179,284)(180,285)(181,286)(182,287)(211,323)(212,324)(213,325)(214,326)(215,327)(216,328)(217,329)(218,330)(219,331)(220,332)(221,333)(222,334)(223,335)(224,336)(225,411)(226,412)(227,413)(228,414)(229,415)(230,416)(231,417)(232,418)(233,419)(234,420)(235,407)(236,408)(237,409)(238,410)(239,439)(240,440)(241,441)(242,442)(243,443)(244,444)(245,445)(246,446)(247,447)(248,448)(249,435)(250,436)(251,437)(252,438)(295,317)(296,318)(297,319)(298,320)(299,321)(300,322)(301,309)(302,310)(303,311)(304,312)(305,313)(306,314)(307,315)(308,316)(337,397)(338,398)(339,399)(340,400)(341,401)(342,402)(343,403)(344,404)(345,405)(346,406)(347,393)(348,394)(349,395)(350,396), (1,15,417,64,42,267,92,158)(2,16,418,65,29,268,93,159)(3,17,419,66,30,269,94,160)(4,18,420,67,31,270,95,161)(5,19,407,68,32,271,96,162)(6,20,408,69,33,272,97,163)(7,21,409,70,34,273,98,164)(8,22,410,57,35,274,85,165)(9,23,411,58,36,275,86,166)(10,24,412,59,37,276,87,167)(11,25,413,60,38,277,88,168)(12,26,414,61,39,278,89,155)(13,27,415,62,40,279,90,156)(14,28,416,63,41,280,91,157)(43,387,234,378,128,74,359,262)(44,388,235,365,129,75,360,263)(45,389,236,366,130,76,361,264)(46,390,237,367,131,77,362,265)(47,391,238,368,132,78,363,266)(48,392,225,369,133,79,364,253)(49,379,226,370,134,80,351,254)(50,380,227,371,135,81,352,255)(51,381,228,372,136,82,353,256)(52,382,229,373,137,83,354,257)(53,383,230,374,138,84,355,258)(54,384,231,375,139,71,356,259)(55,385,232,376,140,72,357,260)(56,386,233,377,127,73,358,261)(99,445,426,282,210,343,310,335)(100,446,427,283,197,344,311,336)(101,447,428,284,198,345,312,323)(102,448,429,285,199,346,313,324)(103,435,430,286,200,347,314,325)(104,436,431,287,201,348,315,326)(105,437,432,288,202,349,316,327)(106,438,433,289,203,350,317,328)(107,439,434,290,204,337,318,329)(108,440,421,291,205,338,319,330)(109,441,422,292,206,339,320,331)(110,442,423,293,207,340,321,332)(111,443,424,294,208,341,322,333)(112,444,425,281,209,342,309,334)(113,172,152,398,297,218,193,240)(114,173,153,399,298,219,194,241)(115,174,154,400,299,220,195,242)(116,175,141,401,300,221,196,243)(117,176,142,402,301,222,183,244)(118,177,143,403,302,223,184,245)(119,178,144,404,303,224,185,246)(120,179,145,405,304,211,186,247)(121,180,146,406,305,212,187,248)(122,181,147,393,306,213,188,249)(123,182,148,394,307,214,189,250)(124,169,149,395,308,215,190,251)(125,170,150,396,295,216,191,252)(126,171,151,397,296,217,192,239), (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), (1,428,8,421)(2,427,9,434)(3,426,10,433)(4,425,11,432)(5,424,12,431)(6,423,13,430)(7,422,14,429)(15,345,22,338)(16,344,23,337)(17,343,24,350)(18,342,25,349)(19,341,26,348)(20,340,27,347)(21,339,28,346)(29,311,36,318)(30,310,37,317)(31,309,38,316)(32,322,39,315)(33,321,40,314)(34,320,41,313)(35,319,42,312)(43,117,50,124)(44,116,51,123)(45,115,52,122)(46,114,53,121)(47,113,54,120)(48,126,55,119)(49,125,56,118)(57,291,64,284)(58,290,65,283)(59,289,66,282)(60,288,67,281)(61,287,68,294)(62,286,69,293)(63,285,70,292)(71,247,78,240)(72,246,79,239)(73,245,80,252)(74,244,81,251)(75,243,82,250)(76,242,83,249)(77,241,84,248)(85,205,92,198)(86,204,93,197)(87,203,94,210)(88,202,95,209)(89,201,96,208)(90,200,97,207)(91,199,98,206)(99,412,106,419)(100,411,107,418)(101,410,108,417)(102,409,109,416)(103,408,110,415)(104,407,111,414)(105,420,112,413)(127,302,134,295)(128,301,135,308)(129,300,136,307)(130,299,137,306)(131,298,138,305)(132,297,139,304)(133,296,140,303)(141,353,148,360)(142,352,149,359)(143,351,150,358)(144,364,151,357)(145,363,152,356)(146,362,153,355)(147,361,154,354)(155,326,162,333)(156,325,163,332)(157,324,164,331)(158,323,165,330)(159,336,166,329)(160,335,167,328)(161,334,168,327)(169,378,176,371)(170,377,177,370)(171,376,178,369)(172,375,179,368)(173,374,180,367)(174,373,181,366)(175,372,182,365)(183,227,190,234)(184,226,191,233)(185,225,192,232)(186,238,193,231)(187,237,194,230)(188,236,195,229)(189,235,196,228)(211,266,218,259)(212,265,219,258)(213,264,220,257)(214,263,221,256)(215,262,222,255)(216,261,223,254)(217,260,224,253)(267,447,274,440)(268,446,275,439)(269,445,276,438)(270,444,277,437)(271,443,278,436)(272,442,279,435)(273,441,280,448)(379,396,386,403)(380,395,387,402)(381,394,388,401)(382,393,389,400)(383,406,390,399)(384,405,391,398)(385,404,392,397) );
G=PermutationGroup([[(1,54),(2,55),(3,56),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,384),(16,385),(17,386),(18,387),(19,388),(20,389),(21,390),(22,391),(23,392),(24,379),(25,380),(26,381),(27,382),(28,383),(29,140),(30,127),(31,128),(32,129),(33,130),(34,131),(35,132),(36,133),(37,134),(38,135),(39,136),(40,137),(41,138),(42,139),(57,368),(58,369),(59,370),(60,371),(61,372),(62,373),(63,374),(64,375),(65,376),(66,377),(67,378),(68,365),(69,366),(70,367),(71,267),(72,268),(73,269),(74,270),(75,271),(76,272),(77,273),(78,274),(79,275),(80,276),(81,277),(82,278),(83,279),(84,280),(85,363),(86,364),(87,351),(88,352),(89,353),(90,354),(91,355),(92,356),(93,357),(94,358),(95,359),(96,360),(97,361),(98,362),(99,184),(100,185),(101,186),(102,187),(103,188),(104,189),(105,190),(106,191),(107,192),(108,193),(109,194),(110,195),(111,196),(112,183),(113,421),(114,422),(115,423),(116,424),(117,425),(118,426),(119,427),(120,428),(121,429),(122,430),(123,431),(124,432),(125,433),(126,434),(141,208),(142,209),(143,210),(144,197),(145,198),(146,199),(147,200),(148,201),(149,202),(150,203),(151,204),(152,205),(153,206),(154,207),(155,256),(156,257),(157,258),(158,259),(159,260),(160,261),(161,262),(162,263),(163,264),(164,265),(165,266),(166,253),(167,254),(168,255),(169,288),(170,289),(171,290),(172,291),(173,292),(174,293),(175,294),(176,281),(177,282),(178,283),(179,284),(180,285),(181,286),(182,287),(211,323),(212,324),(213,325),(214,326),(215,327),(216,328),(217,329),(218,330),(219,331),(220,332),(221,333),(222,334),(223,335),(224,336),(225,411),(226,412),(227,413),(228,414),(229,415),(230,416),(231,417),(232,418),(233,419),(234,420),(235,407),(236,408),(237,409),(238,410),(239,439),(240,440),(241,441),(242,442),(243,443),(244,444),(245,445),(246,446),(247,447),(248,448),(249,435),(250,436),(251,437),(252,438),(295,317),(296,318),(297,319),(298,320),(299,321),(300,322),(301,309),(302,310),(303,311),(304,312),(305,313),(306,314),(307,315),(308,316),(337,397),(338,398),(339,399),(340,400),(341,401),(342,402),(343,403),(344,404),(345,405),(346,406),(347,393),(348,394),(349,395),(350,396)], [(1,15,417,64,42,267,92,158),(2,16,418,65,29,268,93,159),(3,17,419,66,30,269,94,160),(4,18,420,67,31,270,95,161),(5,19,407,68,32,271,96,162),(6,20,408,69,33,272,97,163),(7,21,409,70,34,273,98,164),(8,22,410,57,35,274,85,165),(9,23,411,58,36,275,86,166),(10,24,412,59,37,276,87,167),(11,25,413,60,38,277,88,168),(12,26,414,61,39,278,89,155),(13,27,415,62,40,279,90,156),(14,28,416,63,41,280,91,157),(43,387,234,378,128,74,359,262),(44,388,235,365,129,75,360,263),(45,389,236,366,130,76,361,264),(46,390,237,367,131,77,362,265),(47,391,238,368,132,78,363,266),(48,392,225,369,133,79,364,253),(49,379,226,370,134,80,351,254),(50,380,227,371,135,81,352,255),(51,381,228,372,136,82,353,256),(52,382,229,373,137,83,354,257),(53,383,230,374,138,84,355,258),(54,384,231,375,139,71,356,259),(55,385,232,376,140,72,357,260),(56,386,233,377,127,73,358,261),(99,445,426,282,210,343,310,335),(100,446,427,283,197,344,311,336),(101,447,428,284,198,345,312,323),(102,448,429,285,199,346,313,324),(103,435,430,286,200,347,314,325),(104,436,431,287,201,348,315,326),(105,437,432,288,202,349,316,327),(106,438,433,289,203,350,317,328),(107,439,434,290,204,337,318,329),(108,440,421,291,205,338,319,330),(109,441,422,292,206,339,320,331),(110,442,423,293,207,340,321,332),(111,443,424,294,208,341,322,333),(112,444,425,281,209,342,309,334),(113,172,152,398,297,218,193,240),(114,173,153,399,298,219,194,241),(115,174,154,400,299,220,195,242),(116,175,141,401,300,221,196,243),(117,176,142,402,301,222,183,244),(118,177,143,403,302,223,184,245),(119,178,144,404,303,224,185,246),(120,179,145,405,304,211,186,247),(121,180,146,406,305,212,187,248),(122,181,147,393,306,213,188,249),(123,182,148,394,307,214,189,250),(124,169,149,395,308,215,190,251),(125,170,150,396,295,216,191,252),(126,171,151,397,296,217,192,239)], [(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)], [(1,428,8,421),(2,427,9,434),(3,426,10,433),(4,425,11,432),(5,424,12,431),(6,423,13,430),(7,422,14,429),(15,345,22,338),(16,344,23,337),(17,343,24,350),(18,342,25,349),(19,341,26,348),(20,340,27,347),(21,339,28,346),(29,311,36,318),(30,310,37,317),(31,309,38,316),(32,322,39,315),(33,321,40,314),(34,320,41,313),(35,319,42,312),(43,117,50,124),(44,116,51,123),(45,115,52,122),(46,114,53,121),(47,113,54,120),(48,126,55,119),(49,125,56,118),(57,291,64,284),(58,290,65,283),(59,289,66,282),(60,288,67,281),(61,287,68,294),(62,286,69,293),(63,285,70,292),(71,247,78,240),(72,246,79,239),(73,245,80,252),(74,244,81,251),(75,243,82,250),(76,242,83,249),(77,241,84,248),(85,205,92,198),(86,204,93,197),(87,203,94,210),(88,202,95,209),(89,201,96,208),(90,200,97,207),(91,199,98,206),(99,412,106,419),(100,411,107,418),(101,410,108,417),(102,409,109,416),(103,408,110,415),(104,407,111,414),(105,420,112,413),(127,302,134,295),(128,301,135,308),(129,300,136,307),(130,299,137,306),(131,298,138,305),(132,297,139,304),(133,296,140,303),(141,353,148,360),(142,352,149,359),(143,351,150,358),(144,364,151,357),(145,363,152,356),(146,362,153,355),(147,361,154,354),(155,326,162,333),(156,325,163,332),(157,324,164,331),(158,323,165,330),(159,336,166,329),(160,335,167,328),(161,334,168,327),(169,378,176,371),(170,377,177,370),(171,376,178,369),(172,375,179,368),(173,374,180,367),(174,373,181,366),(175,372,182,365),(183,227,190,234),(184,226,191,233),(185,225,192,232),(186,238,193,231),(187,237,194,230),(188,236,195,229),(189,235,196,228),(211,266,218,259),(212,265,219,258),(213,264,220,257),(214,263,221,256),(215,262,222,255),(216,261,223,254),(217,260,224,253),(267,447,274,440),(268,446,275,439),(269,445,276,438),(270,444,277,437),(271,443,278,436),(272,442,279,435),(273,441,280,448),(379,396,386,403),(380,395,387,402),(381,394,388,401),(382,393,389,400),(383,406,390,399),(384,405,391,398),(385,404,392,397)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D7 | SD16 | Dic7 | D14 | D14 | Dic14 | D28 | D28 | C56⋊C2 |
| kernel | C2×C8⋊Dic7 | C8⋊Dic7 | C2×C4⋊Dic7 | C22×C56 | C2×C56 | C2×C28 | C2×C28 | C22×C14 | C22×C8 | C2×C14 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 8 | 1 | 2 | 1 | 3 | 8 | 12 | 6 | 3 | 12 | 6 | 6 | 48 |
Matrix representation of C2×C8⋊Dic7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 50 | 42 |
| 0 | 0 | 71 | 89 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 104 | 1 |
| 0 | 0 | 112 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 73 | 84 |
| 0 | 0 | 63 | 40 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,50,71,0,0,42,89],[112,0,0,0,0,1,0,0,0,0,104,112,0,0,1,0],[98,0,0,0,0,1,0,0,0,0,73,63,0,0,84,40] >;
C2×C8⋊Dic7 in GAP, Magma, Sage, TeX
C_2\times C_8\rtimes {\rm Dic}_7 % in TeX
G:=Group("C2xC8:Dic7"); // GroupNames label
G:=SmallGroup(448,638);
// by ID
G=gap.SmallGroup(448,638);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,100,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^14=1,d^2=c^7,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations