direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C8×Dic7, C28.25C42, C14⋊2(C4×C8), C56⋊31(C2×C4), (C2×C56)⋊12C4, (C2×C8).344D14, C4.20(C4×Dic7), (C22×C8).18D7, C23.60(C4×D7), C22.15(C8×D7), C14.17(C22×C8), C14.22(C2×C42), (C22×C56).23C2, (C2×C14).24C42, (C4×Dic7).24C4, C28.137(C22×C4), (C2×C28).853C23, (C2×C56).413C22, (C22×C4).460D14, C22.20(C4×Dic7), C4.31(C22×Dic7), (C22×Dic7).19C4, (C22×C28).556C22, (C4×Dic7).302C22, C7⋊3(C2×C4×C8), (C2×C7⋊C8)⋊14C4, C2.3(D7×C2×C8), C7⋊C8⋊29(C2×C4), C4.111(C2×C4×D7), C2.3(C2×C4×Dic7), C22.56(C2×C4×D7), (C2×C14).17(C2×C8), (C2×C4).181(C4×D7), (C22×C7⋊C8).22C2, (C2×C4×Dic7).43C2, (C2×C28).319(C2×C4), (C2×C7⋊C8).333C22, (C22×C14).88(C2×C4), (C2×C4).102(C2×Dic7), (C2×C4).795(C22×D7), (C2×C14).123(C22×C4), (C2×Dic7).110(C2×C4), SmallGroup(448,632)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C8×Dic7 |
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, bd=db, dcd-1=c-1 >
Subgroups: 420 in 162 conjugacy classes, 119 normal (23 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C2×C8, C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C4×C8, C2×C42, C22×C8, C22×C8, C7⋊C8, C56, C2×Dic7, C2×C28, C2×C28, C22×C14, C2×C4×C8, C2×C7⋊C8, C4×Dic7, C2×C56, C22×Dic7, C22×C28, C8×Dic7, C22×C7⋊C8, C2×C4×Dic7, C22×C56, C2×C8×Dic7
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C42, C2×C8, C22×C4, Dic7, D14, C4×C8, C2×C42, C22×C8, C4×D7, C2×Dic7, C22×D7, C2×C4×C8, C8×D7, C4×Dic7, C2×C4×D7, C22×Dic7, C8×Dic7, D7×C2×C8, C2×C4×Dic7, C2×C8×Dic7
►Regular action on 448 points
Generators in S448
(1 290)(2 291)(3 292)(4 293)(5 294)(6 281)(7 282)(8 283)(9 284)(10 285)(11 286)(12 287)(13 288)(14 289)(15 253)(16 254)(17 255)(18 256)(19 257)(20 258)(21 259)(22 260)(23 261)(24 262)(25 263)(26 264)(27 265)(28 266)(29 372)(30 373)(31 374)(32 375)(33 376)(34 377)(35 378)(36 365)(37 366)(38 367)(39 368)(40 369)(41 370)(42 371)(43 220)(44 221)(45 222)(46 223)(47 224)(48 211)(49 212)(50 213)(51 214)(52 215)(53 216)(54 217)(55 218)(56 219)(57 401)(58 402)(59 403)(60 404)(61 405)(62 406)(63 393)(64 394)(65 395)(66 396)(67 397)(68 398)(69 399)(70 400)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 121)(78 122)(79 123)(80 124)(81 125)(82 126)(83 113)(84 114)(85 132)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 127)(95 128)(96 129)(97 130)(98 131)(99 209)(100 210)(101 197)(102 198)(103 199)(104 200)(105 201)(106 202)(107 203)(108 204)(109 205)(110 206)(111 207)(112 208)(141 233)(142 234)(143 235)(144 236)(145 237)(146 238)(147 225)(148 226)(149 227)(150 228)(151 229)(152 230)(153 231)(154 232)(155 186)(156 187)(157 188)(158 189)(159 190)(160 191)(161 192)(162 193)(163 194)(164 195)(165 196)(166 183)(167 184)(168 185)(169 432)(170 433)(171 434)(172 421)(173 422)(174 423)(175 424)(176 425)(177 426)(178 427)(179 428)(180 429)(181 430)(182 431)(239 270)(240 271)(241 272)(242 273)(243 274)(244 275)(245 276)(246 277)(247 278)(248 279)(249 280)(250 267)(251 268)(252 269)(295 311)(296 312)(297 313)(298 314)(299 315)(300 316)(301 317)(302 318)(303 319)(304 320)(305 321)(306 322)(307 309)(308 310)(323 416)(324 417)(325 418)(326 419)(327 420)(328 407)(329 408)(330 409)(331 410)(332 411)(333 412)(334 413)(335 414)(336 415)(337 443)(338 444)(339 445)(340 446)(341 447)(342 448)(343 435)(344 436)(345 437)(346 438)(347 439)(348 440)(349 441)(350 442)(351 383)(352 384)(353 385)(354 386)(355 387)(356 388)(357 389)(358 390)(359 391)(360 392)(361 379)(362 380)(363 381)(364 382)
(1 159 428 272 223 141 312 84)(2 160 429 273 224 142 313 71)(3 161 430 274 211 143 314 72)(4 162 431 275 212 144 315 73)(5 163 432 276 213 145 316 74)(6 164 433 277 214 146 317 75)(7 165 434 278 215 147 318 76)(8 166 421 279 216 148 319 77)(9 167 422 280 217 149 320 78)(10 168 423 267 218 150 321 79)(11 155 424 268 219 151 322 80)(12 156 425 269 220 152 309 81)(13 157 426 270 221 153 310 82)(14 158 427 271 222 154 311 83)(15 101 376 407 70 139 349 390)(16 102 377 408 57 140 350 391)(17 103 378 409 58 127 337 392)(18 104 365 410 59 128 338 379)(19 105 366 411 60 129 339 380)(20 106 367 412 61 130 340 381)(21 107 368 413 62 131 341 382)(22 108 369 414 63 132 342 383)(23 109 370 415 64 133 343 384)(24 110 371 416 65 134 344 385)(25 111 372 417 66 135 345 386)(26 112 373 418 67 136 346 387)(27 99 374 419 68 137 347 388)(28 100 375 420 69 138 348 389)(29 324 396 88 437 354 263 207)(30 325 397 89 438 355 264 208)(31 326 398 90 439 356 265 209)(32 327 399 91 440 357 266 210)(33 328 400 92 441 358 253 197)(34 329 401 93 442 359 254 198)(35 330 402 94 443 360 255 199)(36 331 403 95 444 361 256 200)(37 332 404 96 445 362 257 201)(38 333 405 97 446 363 258 202)(39 334 406 98 447 364 259 203)(40 335 393 85 448 351 260 204)(41 336 394 86 435 352 261 205)(42 323 395 87 436 353 262 206)(43 230 307 125 287 187 176 252)(44 231 308 126 288 188 177 239)(45 232 295 113 289 189 178 240)(46 233 296 114 290 190 179 241)(47 234 297 115 291 191 180 242)(48 235 298 116 292 192 181 243)(49 236 299 117 293 193 182 244)(50 237 300 118 294 194 169 245)(51 238 301 119 281 195 170 246)(52 225 302 120 282 196 171 247)(53 226 303 121 283 183 172 248)(54 227 304 122 284 184 173 249)(55 228 305 123 285 185 174 250)(56 229 306 124 286 186 175 251)
(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)
(1 363 8 356)(2 362 9 355)(3 361 10 354)(4 360 11 353)(5 359 12 352)(6 358 13 351)(7 357 14 364)(15 188 22 195)(16 187 23 194)(17 186 24 193)(18 185 25 192)(19 184 26 191)(20 183 27 190)(21 196 28 189)(29 274 36 267)(30 273 37 280)(31 272 38 279)(32 271 39 278)(33 270 40 277)(34 269 41 276)(35 268 42 275)(43 415 50 408)(44 414 51 407)(45 413 52 420)(46 412 53 419)(47 411 54 418)(48 410 55 417)(49 409 56 416)(57 230 64 237)(58 229 65 236)(59 228 66 235)(60 227 67 234)(61 226 68 233)(62 225 69 232)(63 238 70 231)(71 445 78 438)(72 444 79 437)(73 443 80 436)(74 442 81 435)(75 441 82 448)(76 440 83 447)(77 439 84 446)(85 317 92 310)(86 316 93 309)(87 315 94 322)(88 314 95 321)(89 313 96 320)(90 312 97 319)(91 311 98 318)(99 179 106 172)(100 178 107 171)(101 177 108 170)(102 176 109 169)(103 175 110 182)(104 174 111 181)(105 173 112 180)(113 341 120 348)(114 340 121 347)(115 339 122 346)(116 338 123 345)(117 337 124 344)(118 350 125 343)(119 349 126 342)(127 306 134 299)(128 305 135 298)(129 304 136 297)(130 303 137 296)(131 302 138 295)(132 301 139 308)(133 300 140 307)(141 405 148 398)(142 404 149 397)(143 403 150 396)(144 402 151 395)(145 401 152 394)(146 400 153 393)(147 399 154 406)(155 262 162 255)(156 261 163 254)(157 260 164 253)(158 259 165 266)(159 258 166 265)(160 257 167 264)(161 256 168 263)(197 426 204 433)(198 425 205 432)(199 424 206 431)(200 423 207 430)(201 422 208 429)(202 421 209 428)(203 434 210 427)(211 331 218 324)(212 330 219 323)(213 329 220 336)(214 328 221 335)(215 327 222 334)(216 326 223 333)(217 325 224 332)(239 369 246 376)(240 368 247 375)(241 367 248 374)(242 366 249 373)(243 365 250 372)(244 378 251 371)(245 377 252 370)(281 390 288 383)(282 389 289 382)(283 388 290 381)(284 387 291 380)(285 386 292 379)(286 385 293 392)(287 384 294 391)
G:=sub<Sym(448)| (1,290)(2,291)(3,292)(4,293)(5,294)(6,281)(7,282)(8,283)(9,284)(10,285)(11,286)(12,287)(13,288)(14,289)(15,253)(16,254)(17,255)(18,256)(19,257)(20,258)(21,259)(22,260)(23,261)(24,262)(25,263)(26,264)(27,265)(28,266)(29,372)(30,373)(31,374)(32,375)(33,376)(34,377)(35,378)(36,365)(37,366)(38,367)(39,368)(40,369)(41,370)(42,371)(43,220)(44,221)(45,222)(46,223)(47,224)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(57,401)(58,402)(59,403)(60,404)(61,405)(62,406)(63,393)(64,394)(65,395)(66,396)(67,397)(68,398)(69,399)(70,400)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,121)(78,122)(79,123)(80,124)(81,125)(82,126)(83,113)(84,114)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,127)(95,128)(96,129)(97,130)(98,131)(99,209)(100,210)(101,197)(102,198)(103,199)(104,200)(105,201)(106,202)(107,203)(108,204)(109,205)(110,206)(111,207)(112,208)(141,233)(142,234)(143,235)(144,236)(145,237)(146,238)(147,225)(148,226)(149,227)(150,228)(151,229)(152,230)(153,231)(154,232)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,183)(167,184)(168,185)(169,432)(170,433)(171,434)(172,421)(173,422)(174,423)(175,424)(176,425)(177,426)(178,427)(179,428)(180,429)(181,430)(182,431)(239,270)(240,271)(241,272)(242,273)(243,274)(244,275)(245,276)(246,277)(247,278)(248,279)(249,280)(250,267)(251,268)(252,269)(295,311)(296,312)(297,313)(298,314)(299,315)(300,316)(301,317)(302,318)(303,319)(304,320)(305,321)(306,322)(307,309)(308,310)(323,416)(324,417)(325,418)(326,419)(327,420)(328,407)(329,408)(330,409)(331,410)(332,411)(333,412)(334,413)(335,414)(336,415)(337,443)(338,444)(339,445)(340,446)(341,447)(342,448)(343,435)(344,436)(345,437)(346,438)(347,439)(348,440)(349,441)(350,442)(351,383)(352,384)(353,385)(354,386)(355,387)(356,388)(357,389)(358,390)(359,391)(360,392)(361,379)(362,380)(363,381)(364,382), (1,159,428,272,223,141,312,84)(2,160,429,273,224,142,313,71)(3,161,430,274,211,143,314,72)(4,162,431,275,212,144,315,73)(5,163,432,276,213,145,316,74)(6,164,433,277,214,146,317,75)(7,165,434,278,215,147,318,76)(8,166,421,279,216,148,319,77)(9,167,422,280,217,149,320,78)(10,168,423,267,218,150,321,79)(11,155,424,268,219,151,322,80)(12,156,425,269,220,152,309,81)(13,157,426,270,221,153,310,82)(14,158,427,271,222,154,311,83)(15,101,376,407,70,139,349,390)(16,102,377,408,57,140,350,391)(17,103,378,409,58,127,337,392)(18,104,365,410,59,128,338,379)(19,105,366,411,60,129,339,380)(20,106,367,412,61,130,340,381)(21,107,368,413,62,131,341,382)(22,108,369,414,63,132,342,383)(23,109,370,415,64,133,343,384)(24,110,371,416,65,134,344,385)(25,111,372,417,66,135,345,386)(26,112,373,418,67,136,346,387)(27,99,374,419,68,137,347,388)(28,100,375,420,69,138,348,389)(29,324,396,88,437,354,263,207)(30,325,397,89,438,355,264,208)(31,326,398,90,439,356,265,209)(32,327,399,91,440,357,266,210)(33,328,400,92,441,358,253,197)(34,329,401,93,442,359,254,198)(35,330,402,94,443,360,255,199)(36,331,403,95,444,361,256,200)(37,332,404,96,445,362,257,201)(38,333,405,97,446,363,258,202)(39,334,406,98,447,364,259,203)(40,335,393,85,448,351,260,204)(41,336,394,86,435,352,261,205)(42,323,395,87,436,353,262,206)(43,230,307,125,287,187,176,252)(44,231,308,126,288,188,177,239)(45,232,295,113,289,189,178,240)(46,233,296,114,290,190,179,241)(47,234,297,115,291,191,180,242)(48,235,298,116,292,192,181,243)(49,236,299,117,293,193,182,244)(50,237,300,118,294,194,169,245)(51,238,301,119,281,195,170,246)(52,225,302,120,282,196,171,247)(53,226,303,121,283,183,172,248)(54,227,304,122,284,184,173,249)(55,228,305,123,285,185,174,250)(56,229,306,124,286,186,175,251), (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), (1,363,8,356)(2,362,9,355)(3,361,10,354)(4,360,11,353)(5,359,12,352)(6,358,13,351)(7,357,14,364)(15,188,22,195)(16,187,23,194)(17,186,24,193)(18,185,25,192)(19,184,26,191)(20,183,27,190)(21,196,28,189)(29,274,36,267)(30,273,37,280)(31,272,38,279)(32,271,39,278)(33,270,40,277)(34,269,41,276)(35,268,42,275)(43,415,50,408)(44,414,51,407)(45,413,52,420)(46,412,53,419)(47,411,54,418)(48,410,55,417)(49,409,56,416)(57,230,64,237)(58,229,65,236)(59,228,66,235)(60,227,67,234)(61,226,68,233)(62,225,69,232)(63,238,70,231)(71,445,78,438)(72,444,79,437)(73,443,80,436)(74,442,81,435)(75,441,82,448)(76,440,83,447)(77,439,84,446)(85,317,92,310)(86,316,93,309)(87,315,94,322)(88,314,95,321)(89,313,96,320)(90,312,97,319)(91,311,98,318)(99,179,106,172)(100,178,107,171)(101,177,108,170)(102,176,109,169)(103,175,110,182)(104,174,111,181)(105,173,112,180)(113,341,120,348)(114,340,121,347)(115,339,122,346)(116,338,123,345)(117,337,124,344)(118,350,125,343)(119,349,126,342)(127,306,134,299)(128,305,135,298)(129,304,136,297)(130,303,137,296)(131,302,138,295)(132,301,139,308)(133,300,140,307)(141,405,148,398)(142,404,149,397)(143,403,150,396)(144,402,151,395)(145,401,152,394)(146,400,153,393)(147,399,154,406)(155,262,162,255)(156,261,163,254)(157,260,164,253)(158,259,165,266)(159,258,166,265)(160,257,167,264)(161,256,168,263)(197,426,204,433)(198,425,205,432)(199,424,206,431)(200,423,207,430)(201,422,208,429)(202,421,209,428)(203,434,210,427)(211,331,218,324)(212,330,219,323)(213,329,220,336)(214,328,221,335)(215,327,222,334)(216,326,223,333)(217,325,224,332)(239,369,246,376)(240,368,247,375)(241,367,248,374)(242,366,249,373)(243,365,250,372)(244,378,251,371)(245,377,252,370)(281,390,288,383)(282,389,289,382)(283,388,290,381)(284,387,291,380)(285,386,292,379)(286,385,293,392)(287,384,294,391)>;
G:=Group( (1,290)(2,291)(3,292)(4,293)(5,294)(6,281)(7,282)(8,283)(9,284)(10,285)(11,286)(12,287)(13,288)(14,289)(15,253)(16,254)(17,255)(18,256)(19,257)(20,258)(21,259)(22,260)(23,261)(24,262)(25,263)(26,264)(27,265)(28,266)(29,372)(30,373)(31,374)(32,375)(33,376)(34,377)(35,378)(36,365)(37,366)(38,367)(39,368)(40,369)(41,370)(42,371)(43,220)(44,221)(45,222)(46,223)(47,224)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(57,401)(58,402)(59,403)(60,404)(61,405)(62,406)(63,393)(64,394)(65,395)(66,396)(67,397)(68,398)(69,399)(70,400)(71,115)(72,116)(73,117)(74,118)(75,119)(76,120)(77,121)(78,122)(79,123)(80,124)(81,125)(82,126)(83,113)(84,114)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,127)(95,128)(96,129)(97,130)(98,131)(99,209)(100,210)(101,197)(102,198)(103,199)(104,200)(105,201)(106,202)(107,203)(108,204)(109,205)(110,206)(111,207)(112,208)(141,233)(142,234)(143,235)(144,236)(145,237)(146,238)(147,225)(148,226)(149,227)(150,228)(151,229)(152,230)(153,231)(154,232)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,183)(167,184)(168,185)(169,432)(170,433)(171,434)(172,421)(173,422)(174,423)(175,424)(176,425)(177,426)(178,427)(179,428)(180,429)(181,430)(182,431)(239,270)(240,271)(241,272)(242,273)(243,274)(244,275)(245,276)(246,277)(247,278)(248,279)(249,280)(250,267)(251,268)(252,269)(295,311)(296,312)(297,313)(298,314)(299,315)(300,316)(301,317)(302,318)(303,319)(304,320)(305,321)(306,322)(307,309)(308,310)(323,416)(324,417)(325,418)(326,419)(327,420)(328,407)(329,408)(330,409)(331,410)(332,411)(333,412)(334,413)(335,414)(336,415)(337,443)(338,444)(339,445)(340,446)(341,447)(342,448)(343,435)(344,436)(345,437)(346,438)(347,439)(348,440)(349,441)(350,442)(351,383)(352,384)(353,385)(354,386)(355,387)(356,388)(357,389)(358,390)(359,391)(360,392)(361,379)(362,380)(363,381)(364,382), (1,159,428,272,223,141,312,84)(2,160,429,273,224,142,313,71)(3,161,430,274,211,143,314,72)(4,162,431,275,212,144,315,73)(5,163,432,276,213,145,316,74)(6,164,433,277,214,146,317,75)(7,165,434,278,215,147,318,76)(8,166,421,279,216,148,319,77)(9,167,422,280,217,149,320,78)(10,168,423,267,218,150,321,79)(11,155,424,268,219,151,322,80)(12,156,425,269,220,152,309,81)(13,157,426,270,221,153,310,82)(14,158,427,271,222,154,311,83)(15,101,376,407,70,139,349,390)(16,102,377,408,57,140,350,391)(17,103,378,409,58,127,337,392)(18,104,365,410,59,128,338,379)(19,105,366,411,60,129,339,380)(20,106,367,412,61,130,340,381)(21,107,368,413,62,131,341,382)(22,108,369,414,63,132,342,383)(23,109,370,415,64,133,343,384)(24,110,371,416,65,134,344,385)(25,111,372,417,66,135,345,386)(26,112,373,418,67,136,346,387)(27,99,374,419,68,137,347,388)(28,100,375,420,69,138,348,389)(29,324,396,88,437,354,263,207)(30,325,397,89,438,355,264,208)(31,326,398,90,439,356,265,209)(32,327,399,91,440,357,266,210)(33,328,400,92,441,358,253,197)(34,329,401,93,442,359,254,198)(35,330,402,94,443,360,255,199)(36,331,403,95,444,361,256,200)(37,332,404,96,445,362,257,201)(38,333,405,97,446,363,258,202)(39,334,406,98,447,364,259,203)(40,335,393,85,448,351,260,204)(41,336,394,86,435,352,261,205)(42,323,395,87,436,353,262,206)(43,230,307,125,287,187,176,252)(44,231,308,126,288,188,177,239)(45,232,295,113,289,189,178,240)(46,233,296,114,290,190,179,241)(47,234,297,115,291,191,180,242)(48,235,298,116,292,192,181,243)(49,236,299,117,293,193,182,244)(50,237,300,118,294,194,169,245)(51,238,301,119,281,195,170,246)(52,225,302,120,282,196,171,247)(53,226,303,121,283,183,172,248)(54,227,304,122,284,184,173,249)(55,228,305,123,285,185,174,250)(56,229,306,124,286,186,175,251), (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), (1,363,8,356)(2,362,9,355)(3,361,10,354)(4,360,11,353)(5,359,12,352)(6,358,13,351)(7,357,14,364)(15,188,22,195)(16,187,23,194)(17,186,24,193)(18,185,25,192)(19,184,26,191)(20,183,27,190)(21,196,28,189)(29,274,36,267)(30,273,37,280)(31,272,38,279)(32,271,39,278)(33,270,40,277)(34,269,41,276)(35,268,42,275)(43,415,50,408)(44,414,51,407)(45,413,52,420)(46,412,53,419)(47,411,54,418)(48,410,55,417)(49,409,56,416)(57,230,64,237)(58,229,65,236)(59,228,66,235)(60,227,67,234)(61,226,68,233)(62,225,69,232)(63,238,70,231)(71,445,78,438)(72,444,79,437)(73,443,80,436)(74,442,81,435)(75,441,82,448)(76,440,83,447)(77,439,84,446)(85,317,92,310)(86,316,93,309)(87,315,94,322)(88,314,95,321)(89,313,96,320)(90,312,97,319)(91,311,98,318)(99,179,106,172)(100,178,107,171)(101,177,108,170)(102,176,109,169)(103,175,110,182)(104,174,111,181)(105,173,112,180)(113,341,120,348)(114,340,121,347)(115,339,122,346)(116,338,123,345)(117,337,124,344)(118,350,125,343)(119,349,126,342)(127,306,134,299)(128,305,135,298)(129,304,136,297)(130,303,137,296)(131,302,138,295)(132,301,139,308)(133,300,140,307)(141,405,148,398)(142,404,149,397)(143,403,150,396)(144,402,151,395)(145,401,152,394)(146,400,153,393)(147,399,154,406)(155,262,162,255)(156,261,163,254)(157,260,164,253)(158,259,165,266)(159,258,166,265)(160,257,167,264)(161,256,168,263)(197,426,204,433)(198,425,205,432)(199,424,206,431)(200,423,207,430)(201,422,208,429)(202,421,209,428)(203,434,210,427)(211,331,218,324)(212,330,219,323)(213,329,220,336)(214,328,221,335)(215,327,222,334)(216,326,223,333)(217,325,224,332)(239,369,246,376)(240,368,247,375)(241,367,248,374)(242,366,249,373)(243,365,250,372)(244,378,251,371)(245,377,252,370)(281,390,288,383)(282,389,289,382)(283,388,290,381)(284,387,291,380)(285,386,292,379)(286,385,293,392)(287,384,294,391) );
G=PermutationGroup([[(1,290),(2,291),(3,292),(4,293),(5,294),(6,281),(7,282),(8,283),(9,284),(10,285),(11,286),(12,287),(13,288),(14,289),(15,253),(16,254),(17,255),(18,256),(19,257),(20,258),(21,259),(22,260),(23,261),(24,262),(25,263),(26,264),(27,265),(28,266),(29,372),(30,373),(31,374),(32,375),(33,376),(34,377),(35,378),(36,365),(37,366),(38,367),(39,368),(40,369),(41,370),(42,371),(43,220),(44,221),(45,222),(46,223),(47,224),(48,211),(49,212),(50,213),(51,214),(52,215),(53,216),(54,217),(55,218),(56,219),(57,401),(58,402),(59,403),(60,404),(61,405),(62,406),(63,393),(64,394),(65,395),(66,396),(67,397),(68,398),(69,399),(70,400),(71,115),(72,116),(73,117),(74,118),(75,119),(76,120),(77,121),(78,122),(79,123),(80,124),(81,125),(82,126),(83,113),(84,114),(85,132),(86,133),(87,134),(88,135),(89,136),(90,137),(91,138),(92,139),(93,140),(94,127),(95,128),(96,129),(97,130),(98,131),(99,209),(100,210),(101,197),(102,198),(103,199),(104,200),(105,201),(106,202),(107,203),(108,204),(109,205),(110,206),(111,207),(112,208),(141,233),(142,234),(143,235),(144,236),(145,237),(146,238),(147,225),(148,226),(149,227),(150,228),(151,229),(152,230),(153,231),(154,232),(155,186),(156,187),(157,188),(158,189),(159,190),(160,191),(161,192),(162,193),(163,194),(164,195),(165,196),(166,183),(167,184),(168,185),(169,432),(170,433),(171,434),(172,421),(173,422),(174,423),(175,424),(176,425),(177,426),(178,427),(179,428),(180,429),(181,430),(182,431),(239,270),(240,271),(241,272),(242,273),(243,274),(244,275),(245,276),(246,277),(247,278),(248,279),(249,280),(250,267),(251,268),(252,269),(295,311),(296,312),(297,313),(298,314),(299,315),(300,316),(301,317),(302,318),(303,319),(304,320),(305,321),(306,322),(307,309),(308,310),(323,416),(324,417),(325,418),(326,419),(327,420),(328,407),(329,408),(330,409),(331,410),(332,411),(333,412),(334,413),(335,414),(336,415),(337,443),(338,444),(339,445),(340,446),(341,447),(342,448),(343,435),(344,436),(345,437),(346,438),(347,439),(348,440),(349,441),(350,442),(351,383),(352,384),(353,385),(354,386),(355,387),(356,388),(357,389),(358,390),(359,391),(360,392),(361,379),(362,380),(363,381),(364,382)], [(1,159,428,272,223,141,312,84),(2,160,429,273,224,142,313,71),(3,161,430,274,211,143,314,72),(4,162,431,275,212,144,315,73),(5,163,432,276,213,145,316,74),(6,164,433,277,214,146,317,75),(7,165,434,278,215,147,318,76),(8,166,421,279,216,148,319,77),(9,167,422,280,217,149,320,78),(10,168,423,267,218,150,321,79),(11,155,424,268,219,151,322,80),(12,156,425,269,220,152,309,81),(13,157,426,270,221,153,310,82),(14,158,427,271,222,154,311,83),(15,101,376,407,70,139,349,390),(16,102,377,408,57,140,350,391),(17,103,378,409,58,127,337,392),(18,104,365,410,59,128,338,379),(19,105,366,411,60,129,339,380),(20,106,367,412,61,130,340,381),(21,107,368,413,62,131,341,382),(22,108,369,414,63,132,342,383),(23,109,370,415,64,133,343,384),(24,110,371,416,65,134,344,385),(25,111,372,417,66,135,345,386),(26,112,373,418,67,136,346,387),(27,99,374,419,68,137,347,388),(28,100,375,420,69,138,348,389),(29,324,396,88,437,354,263,207),(30,325,397,89,438,355,264,208),(31,326,398,90,439,356,265,209),(32,327,399,91,440,357,266,210),(33,328,400,92,441,358,253,197),(34,329,401,93,442,359,254,198),(35,330,402,94,443,360,255,199),(36,331,403,95,444,361,256,200),(37,332,404,96,445,362,257,201),(38,333,405,97,446,363,258,202),(39,334,406,98,447,364,259,203),(40,335,393,85,448,351,260,204),(41,336,394,86,435,352,261,205),(42,323,395,87,436,353,262,206),(43,230,307,125,287,187,176,252),(44,231,308,126,288,188,177,239),(45,232,295,113,289,189,178,240),(46,233,296,114,290,190,179,241),(47,234,297,115,291,191,180,242),(48,235,298,116,292,192,181,243),(49,236,299,117,293,193,182,244),(50,237,300,118,294,194,169,245),(51,238,301,119,281,195,170,246),(52,225,302,120,282,196,171,247),(53,226,303,121,283,183,172,248),(54,227,304,122,284,184,173,249),(55,228,305,123,285,185,174,250),(56,229,306,124,286,186,175,251)], [(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)], [(1,363,8,356),(2,362,9,355),(3,361,10,354),(4,360,11,353),(5,359,12,352),(6,358,13,351),(7,357,14,364),(15,188,22,195),(16,187,23,194),(17,186,24,193),(18,185,25,192),(19,184,26,191),(20,183,27,190),(21,196,28,189),(29,274,36,267),(30,273,37,280),(31,272,38,279),(32,271,39,278),(33,270,40,277),(34,269,41,276),(35,268,42,275),(43,415,50,408),(44,414,51,407),(45,413,52,420),(46,412,53,419),(47,411,54,418),(48,410,55,417),(49,409,56,416),(57,230,64,237),(58,229,65,236),(59,228,66,235),(60,227,67,234),(61,226,68,233),(62,225,69,232),(63,238,70,231),(71,445,78,438),(72,444,79,437),(73,443,80,436),(74,442,81,435),(75,441,82,448),(76,440,83,447),(77,439,84,446),(85,317,92,310),(86,316,93,309),(87,315,94,322),(88,314,95,321),(89,313,96,320),(90,312,97,319),(91,311,98,318),(99,179,106,172),(100,178,107,171),(101,177,108,170),(102,176,109,169),(103,175,110,182),(104,174,111,181),(105,173,112,180),(113,341,120,348),(114,340,121,347),(115,339,122,346),(116,338,123,345),(117,337,124,344),(118,350,125,343),(119,349,126,342),(127,306,134,299),(128,305,135,298),(129,304,136,297),(130,303,137,296),(131,302,138,295),(132,301,139,308),(133,300,140,307),(141,405,148,398),(142,404,149,397),(143,403,150,396),(144,402,151,395),(145,401,152,394),(146,400,153,393),(147,399,154,406),(155,262,162,255),(156,261,163,254),(157,260,164,253),(158,259,165,266),(159,258,166,265),(160,257,167,264),(161,256,168,263),(197,426,204,433),(198,425,205,432),(199,424,206,431),(200,423,207,430),(201,422,208,429),(202,421,209,428),(203,434,210,427),(211,331,218,324),(212,330,219,323),(213,329,220,336),(214,328,221,335),(215,327,222,334),(216,326,223,333),(217,325,224,332),(239,369,246,376),(240,368,247,375),(241,367,248,374),(242,366,249,373),(243,365,250,372),(244,378,251,371),(245,377,252,370),(281,390,288,383),(282,389,289,382),(283,388,290,381),(284,387,291,380),(285,386,292,379),(286,385,293,392),(287,384,294,391)]])
160 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4X | 7A | 7B | 7C | 8A | ··· | 8P | 8Q | ··· | 8AF | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | D7 | Dic7 | D14 | D14 | C4×D7 | C4×D7 | C8×D7 |
| kernel | C2×C8×Dic7 | C8×Dic7 | C22×C7⋊C8 | C2×C4×Dic7 | C22×C56 | C2×C7⋊C8 | C4×Dic7 | C2×C56 | C22×Dic7 | C2×Dic7 | C22×C8 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 8 | 4 | 32 | 3 | 12 | 6 | 3 | 18 | 6 | 48 |
Matrix representation of C2×C8×Dic7 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 69 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 112 | 9 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 34 | 101 |
| 0 | 0 | 0 | 68 | 79 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1],[69,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,1,9],[1,0,0,0,0,0,98,0,0,0,0,0,112,0,0,0,0,0,34,68,0,0,0,101,79] >;
C2×C8×Dic7 in GAP, Magma, Sage, TeX
C_2\times C_8\times {\rm Dic}_7 % in TeX
G:=Group("C2xC8xDic7"); // GroupNames label
G:=SmallGroup(448,632);
// by ID
G=gap.SmallGroup(448,632);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,100,102,18822]);
// 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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations