direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4×Dic14, C42.271D14, C14⋊1(C4×Q8), C28⋊6(C2×Q8), (C2×C28)⋊14Q8, C14.1(C23×C4), (C2×C42).18D7, C14.1(C22×Q8), (C2×C14).11C24, (C4×C28).332C22, C28.117(C22×C4), (C2×C28).873C23, (C22×C4).465D14, Dic7.1(C22×C4), C2.1(C22×Dic14), C22.11(C23×D7), C22.65(C4○D28), C4⋊Dic7.394C22, C22.33(C2×Dic14), C23.309(C22×D7), Dic7⋊C4.173C22, (C22×C28).500C22, (C22×C14).373C23, (C22×Dic14).20C2, (C4×Dic7).287C22, (C2×Dic7).171C23, (C2×Dic14).313C22, (C22×Dic7).199C22, C7⋊1(C2×C4×Q8), C4.75(C2×C4×D7), (C2×C4×C28).22C2, C2.4(D7×C22×C4), C2.1(C2×C4○D28), C14.1(C2×C4○D4), C22.66(C2×C4×D7), (C2×C4).118(C4×D7), (C2×C14).45(C2×Q8), (C2×C4×Dic7).39C2, (C2×C28).230(C2×C4), (C2×C4⋊Dic7).47C2, (C2×C14).93(C4○D4), (C2×Dic7⋊C4).36C2, (C2×Dic7).69(C2×C4), (C2×C4).815(C22×D7), (C2×C14).144(C22×C4), SmallGroup(448,920)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4×Dic14
G = < a,b,c,d | a2=b4=c28=1, d2=c14, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 964 in 298 conjugacy classes, 183 normal (23 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C2×C14, C2×C42, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C2×C4×Q8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×Dic14, C22×Dic7, C22×C28, C4×Dic14, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C2×C4×C28, C22×Dic14, C2×C4×Dic14
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D7, C22×C4, C2×Q8, C4○D4, C24, D14, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, Dic14, C4×D7, C22×D7, C2×C4×Q8, C2×Dic14, C2×C4×D7, C4○D28, C23×D7, C4×Dic14, C22×Dic14, D7×C22×C4, C2×C4○D28, C2×C4×Dic14
►Regular action on 448 points
Generators in S448
(1 328)(2 329)(3 330)(4 331)(5 332)(6 333)(7 334)(8 335)(9 336)(10 309)(11 310)(12 311)(13 312)(14 313)(15 314)(16 315)(17 316)(18 317)(19 318)(20 319)(21 320)(22 321)(23 322)(24 323)(25 324)(26 325)(27 326)(28 327)(29 223)(30 224)(31 197)(32 198)(33 199)(34 200)(35 201)(36 202)(37 203)(38 204)(39 205)(40 206)(41 207)(42 208)(43 209)(44 210)(45 211)(46 212)(47 213)(48 214)(49 215)(50 216)(51 217)(52 218)(53 219)(54 220)(55 221)(56 222)(57 102)(58 103)(59 104)(60 105)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(113 382)(114 383)(115 384)(116 385)(117 386)(118 387)(119 388)(120 389)(121 390)(122 391)(123 392)(124 365)(125 366)(126 367)(127 368)(128 369)(129 370)(130 371)(131 372)(132 373)(133 374)(134 375)(135 376)(136 377)(137 378)(138 379)(139 380)(140 381)(141 263)(142 264)(143 265)(144 266)(145 267)(146 268)(147 269)(148 270)(149 271)(150 272)(151 273)(152 274)(153 275)(154 276)(155 277)(156 278)(157 279)(158 280)(159 253)(160 254)(161 255)(162 256)(163 257)(164 258)(165 259)(166 260)(167 261)(168 262)(169 227)(170 228)(171 229)(172 230)(173 231)(174 232)(175 233)(176 234)(177 235)(178 236)(179 237)(180 238)(181 239)(182 240)(183 241)(184 242)(185 243)(186 244)(187 245)(188 246)(189 247)(190 248)(191 249)(192 250)(193 251)(194 252)(195 225)(196 226)(281 418)(282 419)(283 420)(284 393)(285 394)(286 395)(287 396)(288 397)(289 398)(290 399)(291 400)(292 401)(293 402)(294 403)(295 404)(296 405)(297 406)(298 407)(299 408)(300 409)(301 410)(302 411)(303 412)(304 413)(305 414)(306 415)(307 416)(308 417)(337 433)(338 434)(339 435)(340 436)(341 437)(342 438)(343 439)(344 440)(345 441)(346 442)(347 443)(348 444)(349 445)(350 446)(351 447)(352 448)(353 421)(354 422)(355 423)(356 424)(357 425)(358 426)(359 427)(360 428)(361 429)(362 430)(363 431)(364 432)
(1 280 436 216)(2 253 437 217)(3 254 438 218)(4 255 439 219)(5 256 440 220)(6 257 441 221)(7 258 442 222)(8 259 443 223)(9 260 444 224)(10 261 445 197)(11 262 446 198)(12 263 447 199)(13 264 448 200)(14 265 421 201)(15 266 422 202)(16 267 423 203)(17 268 424 204)(18 269 425 205)(19 270 426 206)(20 271 427 207)(21 272 428 208)(22 273 429 209)(23 274 430 210)(24 275 431 211)(25 276 432 212)(26 277 433 213)(27 278 434 214)(28 279 435 215)(29 335 165 347)(30 336 166 348)(31 309 167 349)(32 310 168 350)(33 311 141 351)(34 312 142 352)(35 313 143 353)(36 314 144 354)(37 315 145 355)(38 316 146 356)(39 317 147 357)(40 318 148 358)(41 319 149 359)(42 320 150 360)(43 321 151 361)(44 322 152 362)(45 323 153 363)(46 324 154 364)(47 325 155 337)(48 326 156 338)(49 327 157 339)(50 328 158 340)(51 329 159 341)(52 330 160 342)(53 331 161 343)(54 332 162 344)(55 333 163 345)(56 334 164 346)(57 180 376 301)(58 181 377 302)(59 182 378 303)(60 183 379 304)(61 184 380 305)(62 185 381 306)(63 186 382 307)(64 187 383 308)(65 188 384 281)(66 189 385 282)(67 190 386 283)(68 191 387 284)(69 192 388 285)(70 193 389 286)(71 194 390 287)(72 195 391 288)(73 196 392 289)(74 169 365 290)(75 170 366 291)(76 171 367 292)(77 172 368 293)(78 173 369 294)(79 174 370 295)(80 175 371 296)(81 176 372 297)(82 177 373 298)(83 178 374 299)(84 179 375 300)(85 249 118 393)(86 250 119 394)(87 251 120 395)(88 252 121 396)(89 225 122 397)(90 226 123 398)(91 227 124 399)(92 228 125 400)(93 229 126 401)(94 230 127 402)(95 231 128 403)(96 232 129 404)(97 233 130 405)(98 234 131 406)(99 235 132 407)(100 236 133 408)(101 237 134 409)(102 238 135 410)(103 239 136 411)(104 240 137 412)(105 241 138 413)(106 242 139 414)(107 243 140 415)(108 244 113 416)(109 245 114 417)(110 246 115 418)(111 247 116 419)(112 248 117 420)
(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)
(1 170 15 184)(2 169 16 183)(3 196 17 182)(4 195 18 181)(5 194 19 180)(6 193 20 179)(7 192 21 178)(8 191 22 177)(9 190 23 176)(10 189 24 175)(11 188 25 174)(12 187 26 173)(13 186 27 172)(14 185 28 171)(29 85 43 99)(30 112 44 98)(31 111 45 97)(32 110 46 96)(33 109 47 95)(34 108 48 94)(35 107 49 93)(36 106 50 92)(37 105 51 91)(38 104 52 90)(39 103 53 89)(40 102 54 88)(41 101 55 87)(42 100 56 86)(57 220 71 206)(58 219 72 205)(59 218 73 204)(60 217 74 203)(61 216 75 202)(62 215 76 201)(63 214 77 200)(64 213 78 199)(65 212 79 198)(66 211 80 197)(67 210 81 224)(68 209 82 223)(69 208 83 222)(70 207 84 221)(113 156 127 142)(114 155 128 141)(115 154 129 168)(116 153 130 167)(117 152 131 166)(118 151 132 165)(119 150 133 164)(120 149 134 163)(121 148 135 162)(122 147 136 161)(123 146 137 160)(124 145 138 159)(125 144 139 158)(126 143 140 157)(225 317 239 331)(226 316 240 330)(227 315 241 329)(228 314 242 328)(229 313 243 327)(230 312 244 326)(231 311 245 325)(232 310 246 324)(233 309 247 323)(234 336 248 322)(235 335 249 321)(236 334 250 320)(237 333 251 319)(238 332 252 318)(253 365 267 379)(254 392 268 378)(255 391 269 377)(256 390 270 376)(257 389 271 375)(258 388 272 374)(259 387 273 373)(260 386 274 372)(261 385 275 371)(262 384 276 370)(263 383 277 369)(264 382 278 368)(265 381 279 367)(266 380 280 366)(281 432 295 446)(282 431 296 445)(283 430 297 444)(284 429 298 443)(285 428 299 442)(286 427 300 441)(287 426 301 440)(288 425 302 439)(289 424 303 438)(290 423 304 437)(291 422 305 436)(292 421 306 435)(293 448 307 434)(294 447 308 433)(337 403 351 417)(338 402 352 416)(339 401 353 415)(340 400 354 414)(341 399 355 413)(342 398 356 412)(343 397 357 411)(344 396 358 410)(345 395 359 409)(346 394 360 408)(347 393 361 407)(348 420 362 406)(349 419 363 405)(350 418 364 404)
G:=sub<Sym(448)| (1,328)(2,329)(3,330)(4,331)(5,332)(6,333)(7,334)(8,335)(9,336)(10,309)(11,310)(12,311)(13,312)(14,313)(15,314)(16,315)(17,316)(18,317)(19,318)(20,319)(21,320)(22,321)(23,322)(24,323)(25,324)(26,325)(27,326)(28,327)(29,223)(30,224)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,209)(44,210)(45,211)(46,212)(47,213)(48,214)(49,215)(50,216)(51,217)(52,218)(53,219)(54,220)(55,221)(56,222)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(113,382)(114,383)(115,384)(116,385)(117,386)(118,387)(119,388)(120,389)(121,390)(122,391)(123,392)(124,365)(125,366)(126,367)(127,368)(128,369)(129,370)(130,371)(131,372)(132,373)(133,374)(134,375)(135,376)(136,377)(137,378)(138,379)(139,380)(140,381)(141,263)(142,264)(143,265)(144,266)(145,267)(146,268)(147,269)(148,270)(149,271)(150,272)(151,273)(152,274)(153,275)(154,276)(155,277)(156,278)(157,279)(158,280)(159,253)(160,254)(161,255)(162,256)(163,257)(164,258)(165,259)(166,260)(167,261)(168,262)(169,227)(170,228)(171,229)(172,230)(173,231)(174,232)(175,233)(176,234)(177,235)(178,236)(179,237)(180,238)(181,239)(182,240)(183,241)(184,242)(185,243)(186,244)(187,245)(188,246)(189,247)(190,248)(191,249)(192,250)(193,251)(194,252)(195,225)(196,226)(281,418)(282,419)(283,420)(284,393)(285,394)(286,395)(287,396)(288,397)(289,398)(290,399)(291,400)(292,401)(293,402)(294,403)(295,404)(296,405)(297,406)(298,407)(299,408)(300,409)(301,410)(302,411)(303,412)(304,413)(305,414)(306,415)(307,416)(308,417)(337,433)(338,434)(339,435)(340,436)(341,437)(342,438)(343,439)(344,440)(345,441)(346,442)(347,443)(348,444)(349,445)(350,446)(351,447)(352,448)(353,421)(354,422)(355,423)(356,424)(357,425)(358,426)(359,427)(360,428)(361,429)(362,430)(363,431)(364,432), (1,280,436,216)(2,253,437,217)(3,254,438,218)(4,255,439,219)(5,256,440,220)(6,257,441,221)(7,258,442,222)(8,259,443,223)(9,260,444,224)(10,261,445,197)(11,262,446,198)(12,263,447,199)(13,264,448,200)(14,265,421,201)(15,266,422,202)(16,267,423,203)(17,268,424,204)(18,269,425,205)(19,270,426,206)(20,271,427,207)(21,272,428,208)(22,273,429,209)(23,274,430,210)(24,275,431,211)(25,276,432,212)(26,277,433,213)(27,278,434,214)(28,279,435,215)(29,335,165,347)(30,336,166,348)(31,309,167,349)(32,310,168,350)(33,311,141,351)(34,312,142,352)(35,313,143,353)(36,314,144,354)(37,315,145,355)(38,316,146,356)(39,317,147,357)(40,318,148,358)(41,319,149,359)(42,320,150,360)(43,321,151,361)(44,322,152,362)(45,323,153,363)(46,324,154,364)(47,325,155,337)(48,326,156,338)(49,327,157,339)(50,328,158,340)(51,329,159,341)(52,330,160,342)(53,331,161,343)(54,332,162,344)(55,333,163,345)(56,334,164,346)(57,180,376,301)(58,181,377,302)(59,182,378,303)(60,183,379,304)(61,184,380,305)(62,185,381,306)(63,186,382,307)(64,187,383,308)(65,188,384,281)(66,189,385,282)(67,190,386,283)(68,191,387,284)(69,192,388,285)(70,193,389,286)(71,194,390,287)(72,195,391,288)(73,196,392,289)(74,169,365,290)(75,170,366,291)(76,171,367,292)(77,172,368,293)(78,173,369,294)(79,174,370,295)(80,175,371,296)(81,176,372,297)(82,177,373,298)(83,178,374,299)(84,179,375,300)(85,249,118,393)(86,250,119,394)(87,251,120,395)(88,252,121,396)(89,225,122,397)(90,226,123,398)(91,227,124,399)(92,228,125,400)(93,229,126,401)(94,230,127,402)(95,231,128,403)(96,232,129,404)(97,233,130,405)(98,234,131,406)(99,235,132,407)(100,236,133,408)(101,237,134,409)(102,238,135,410)(103,239,136,411)(104,240,137,412)(105,241,138,413)(106,242,139,414)(107,243,140,415)(108,244,113,416)(109,245,114,417)(110,246,115,418)(111,247,116,419)(112,248,117,420), (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), (1,170,15,184)(2,169,16,183)(3,196,17,182)(4,195,18,181)(5,194,19,180)(6,193,20,179)(7,192,21,178)(8,191,22,177)(9,190,23,176)(10,189,24,175)(11,188,25,174)(12,187,26,173)(13,186,27,172)(14,185,28,171)(29,85,43,99)(30,112,44,98)(31,111,45,97)(32,110,46,96)(33,109,47,95)(34,108,48,94)(35,107,49,93)(36,106,50,92)(37,105,51,91)(38,104,52,90)(39,103,53,89)(40,102,54,88)(41,101,55,87)(42,100,56,86)(57,220,71,206)(58,219,72,205)(59,218,73,204)(60,217,74,203)(61,216,75,202)(62,215,76,201)(63,214,77,200)(64,213,78,199)(65,212,79,198)(66,211,80,197)(67,210,81,224)(68,209,82,223)(69,208,83,222)(70,207,84,221)(113,156,127,142)(114,155,128,141)(115,154,129,168)(116,153,130,167)(117,152,131,166)(118,151,132,165)(119,150,133,164)(120,149,134,163)(121,148,135,162)(122,147,136,161)(123,146,137,160)(124,145,138,159)(125,144,139,158)(126,143,140,157)(225,317,239,331)(226,316,240,330)(227,315,241,329)(228,314,242,328)(229,313,243,327)(230,312,244,326)(231,311,245,325)(232,310,246,324)(233,309,247,323)(234,336,248,322)(235,335,249,321)(236,334,250,320)(237,333,251,319)(238,332,252,318)(253,365,267,379)(254,392,268,378)(255,391,269,377)(256,390,270,376)(257,389,271,375)(258,388,272,374)(259,387,273,373)(260,386,274,372)(261,385,275,371)(262,384,276,370)(263,383,277,369)(264,382,278,368)(265,381,279,367)(266,380,280,366)(281,432,295,446)(282,431,296,445)(283,430,297,444)(284,429,298,443)(285,428,299,442)(286,427,300,441)(287,426,301,440)(288,425,302,439)(289,424,303,438)(290,423,304,437)(291,422,305,436)(292,421,306,435)(293,448,307,434)(294,447,308,433)(337,403,351,417)(338,402,352,416)(339,401,353,415)(340,400,354,414)(341,399,355,413)(342,398,356,412)(343,397,357,411)(344,396,358,410)(345,395,359,409)(346,394,360,408)(347,393,361,407)(348,420,362,406)(349,419,363,405)(350,418,364,404)>;
G:=Group( (1,328)(2,329)(3,330)(4,331)(5,332)(6,333)(7,334)(8,335)(9,336)(10,309)(11,310)(12,311)(13,312)(14,313)(15,314)(16,315)(17,316)(18,317)(19,318)(20,319)(21,320)(22,321)(23,322)(24,323)(25,324)(26,325)(27,326)(28,327)(29,223)(30,224)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,209)(44,210)(45,211)(46,212)(47,213)(48,214)(49,215)(50,216)(51,217)(52,218)(53,219)(54,220)(55,221)(56,222)(57,102)(58,103)(59,104)(60,105)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(113,382)(114,383)(115,384)(116,385)(117,386)(118,387)(119,388)(120,389)(121,390)(122,391)(123,392)(124,365)(125,366)(126,367)(127,368)(128,369)(129,370)(130,371)(131,372)(132,373)(133,374)(134,375)(135,376)(136,377)(137,378)(138,379)(139,380)(140,381)(141,263)(142,264)(143,265)(144,266)(145,267)(146,268)(147,269)(148,270)(149,271)(150,272)(151,273)(152,274)(153,275)(154,276)(155,277)(156,278)(157,279)(158,280)(159,253)(160,254)(161,255)(162,256)(163,257)(164,258)(165,259)(166,260)(167,261)(168,262)(169,227)(170,228)(171,229)(172,230)(173,231)(174,232)(175,233)(176,234)(177,235)(178,236)(179,237)(180,238)(181,239)(182,240)(183,241)(184,242)(185,243)(186,244)(187,245)(188,246)(189,247)(190,248)(191,249)(192,250)(193,251)(194,252)(195,225)(196,226)(281,418)(282,419)(283,420)(284,393)(285,394)(286,395)(287,396)(288,397)(289,398)(290,399)(291,400)(292,401)(293,402)(294,403)(295,404)(296,405)(297,406)(298,407)(299,408)(300,409)(301,410)(302,411)(303,412)(304,413)(305,414)(306,415)(307,416)(308,417)(337,433)(338,434)(339,435)(340,436)(341,437)(342,438)(343,439)(344,440)(345,441)(346,442)(347,443)(348,444)(349,445)(350,446)(351,447)(352,448)(353,421)(354,422)(355,423)(356,424)(357,425)(358,426)(359,427)(360,428)(361,429)(362,430)(363,431)(364,432), (1,280,436,216)(2,253,437,217)(3,254,438,218)(4,255,439,219)(5,256,440,220)(6,257,441,221)(7,258,442,222)(8,259,443,223)(9,260,444,224)(10,261,445,197)(11,262,446,198)(12,263,447,199)(13,264,448,200)(14,265,421,201)(15,266,422,202)(16,267,423,203)(17,268,424,204)(18,269,425,205)(19,270,426,206)(20,271,427,207)(21,272,428,208)(22,273,429,209)(23,274,430,210)(24,275,431,211)(25,276,432,212)(26,277,433,213)(27,278,434,214)(28,279,435,215)(29,335,165,347)(30,336,166,348)(31,309,167,349)(32,310,168,350)(33,311,141,351)(34,312,142,352)(35,313,143,353)(36,314,144,354)(37,315,145,355)(38,316,146,356)(39,317,147,357)(40,318,148,358)(41,319,149,359)(42,320,150,360)(43,321,151,361)(44,322,152,362)(45,323,153,363)(46,324,154,364)(47,325,155,337)(48,326,156,338)(49,327,157,339)(50,328,158,340)(51,329,159,341)(52,330,160,342)(53,331,161,343)(54,332,162,344)(55,333,163,345)(56,334,164,346)(57,180,376,301)(58,181,377,302)(59,182,378,303)(60,183,379,304)(61,184,380,305)(62,185,381,306)(63,186,382,307)(64,187,383,308)(65,188,384,281)(66,189,385,282)(67,190,386,283)(68,191,387,284)(69,192,388,285)(70,193,389,286)(71,194,390,287)(72,195,391,288)(73,196,392,289)(74,169,365,290)(75,170,366,291)(76,171,367,292)(77,172,368,293)(78,173,369,294)(79,174,370,295)(80,175,371,296)(81,176,372,297)(82,177,373,298)(83,178,374,299)(84,179,375,300)(85,249,118,393)(86,250,119,394)(87,251,120,395)(88,252,121,396)(89,225,122,397)(90,226,123,398)(91,227,124,399)(92,228,125,400)(93,229,126,401)(94,230,127,402)(95,231,128,403)(96,232,129,404)(97,233,130,405)(98,234,131,406)(99,235,132,407)(100,236,133,408)(101,237,134,409)(102,238,135,410)(103,239,136,411)(104,240,137,412)(105,241,138,413)(106,242,139,414)(107,243,140,415)(108,244,113,416)(109,245,114,417)(110,246,115,418)(111,247,116,419)(112,248,117,420), (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), (1,170,15,184)(2,169,16,183)(3,196,17,182)(4,195,18,181)(5,194,19,180)(6,193,20,179)(7,192,21,178)(8,191,22,177)(9,190,23,176)(10,189,24,175)(11,188,25,174)(12,187,26,173)(13,186,27,172)(14,185,28,171)(29,85,43,99)(30,112,44,98)(31,111,45,97)(32,110,46,96)(33,109,47,95)(34,108,48,94)(35,107,49,93)(36,106,50,92)(37,105,51,91)(38,104,52,90)(39,103,53,89)(40,102,54,88)(41,101,55,87)(42,100,56,86)(57,220,71,206)(58,219,72,205)(59,218,73,204)(60,217,74,203)(61,216,75,202)(62,215,76,201)(63,214,77,200)(64,213,78,199)(65,212,79,198)(66,211,80,197)(67,210,81,224)(68,209,82,223)(69,208,83,222)(70,207,84,221)(113,156,127,142)(114,155,128,141)(115,154,129,168)(116,153,130,167)(117,152,131,166)(118,151,132,165)(119,150,133,164)(120,149,134,163)(121,148,135,162)(122,147,136,161)(123,146,137,160)(124,145,138,159)(125,144,139,158)(126,143,140,157)(225,317,239,331)(226,316,240,330)(227,315,241,329)(228,314,242,328)(229,313,243,327)(230,312,244,326)(231,311,245,325)(232,310,246,324)(233,309,247,323)(234,336,248,322)(235,335,249,321)(236,334,250,320)(237,333,251,319)(238,332,252,318)(253,365,267,379)(254,392,268,378)(255,391,269,377)(256,390,270,376)(257,389,271,375)(258,388,272,374)(259,387,273,373)(260,386,274,372)(261,385,275,371)(262,384,276,370)(263,383,277,369)(264,382,278,368)(265,381,279,367)(266,380,280,366)(281,432,295,446)(282,431,296,445)(283,430,297,444)(284,429,298,443)(285,428,299,442)(286,427,300,441)(287,426,301,440)(288,425,302,439)(289,424,303,438)(290,423,304,437)(291,422,305,436)(292,421,306,435)(293,448,307,434)(294,447,308,433)(337,403,351,417)(338,402,352,416)(339,401,353,415)(340,400,354,414)(341,399,355,413)(342,398,356,412)(343,397,357,411)(344,396,358,410)(345,395,359,409)(346,394,360,408)(347,393,361,407)(348,420,362,406)(349,419,363,405)(350,418,364,404) );
G=PermutationGroup([[(1,328),(2,329),(3,330),(4,331),(5,332),(6,333),(7,334),(8,335),(9,336),(10,309),(11,310),(12,311),(13,312),(14,313),(15,314),(16,315),(17,316),(18,317),(19,318),(20,319),(21,320),(22,321),(23,322),(24,323),(25,324),(26,325),(27,326),(28,327),(29,223),(30,224),(31,197),(32,198),(33,199),(34,200),(35,201),(36,202),(37,203),(38,204),(39,205),(40,206),(41,207),(42,208),(43,209),(44,210),(45,211),(46,212),(47,213),(48,214),(49,215),(50,216),(51,217),(52,218),(53,219),(54,220),(55,221),(56,222),(57,102),(58,103),(59,104),(60,105),(61,106),(62,107),(63,108),(64,109),(65,110),(66,111),(67,112),(68,85),(69,86),(70,87),(71,88),(72,89),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,97),(81,98),(82,99),(83,100),(84,101),(113,382),(114,383),(115,384),(116,385),(117,386),(118,387),(119,388),(120,389),(121,390),(122,391),(123,392),(124,365),(125,366),(126,367),(127,368),(128,369),(129,370),(130,371),(131,372),(132,373),(133,374),(134,375),(135,376),(136,377),(137,378),(138,379),(139,380),(140,381),(141,263),(142,264),(143,265),(144,266),(145,267),(146,268),(147,269),(148,270),(149,271),(150,272),(151,273),(152,274),(153,275),(154,276),(155,277),(156,278),(157,279),(158,280),(159,253),(160,254),(161,255),(162,256),(163,257),(164,258),(165,259),(166,260),(167,261),(168,262),(169,227),(170,228),(171,229),(172,230),(173,231),(174,232),(175,233),(176,234),(177,235),(178,236),(179,237),(180,238),(181,239),(182,240),(183,241),(184,242),(185,243),(186,244),(187,245),(188,246),(189,247),(190,248),(191,249),(192,250),(193,251),(194,252),(195,225),(196,226),(281,418),(282,419),(283,420),(284,393),(285,394),(286,395),(287,396),(288,397),(289,398),(290,399),(291,400),(292,401),(293,402),(294,403),(295,404),(296,405),(297,406),(298,407),(299,408),(300,409),(301,410),(302,411),(303,412),(304,413),(305,414),(306,415),(307,416),(308,417),(337,433),(338,434),(339,435),(340,436),(341,437),(342,438),(343,439),(344,440),(345,441),(346,442),(347,443),(348,444),(349,445),(350,446),(351,447),(352,448),(353,421),(354,422),(355,423),(356,424),(357,425),(358,426),(359,427),(360,428),(361,429),(362,430),(363,431),(364,432)], [(1,280,436,216),(2,253,437,217),(3,254,438,218),(4,255,439,219),(5,256,440,220),(6,257,441,221),(7,258,442,222),(8,259,443,223),(9,260,444,224),(10,261,445,197),(11,262,446,198),(12,263,447,199),(13,264,448,200),(14,265,421,201),(15,266,422,202),(16,267,423,203),(17,268,424,204),(18,269,425,205),(19,270,426,206),(20,271,427,207),(21,272,428,208),(22,273,429,209),(23,274,430,210),(24,275,431,211),(25,276,432,212),(26,277,433,213),(27,278,434,214),(28,279,435,215),(29,335,165,347),(30,336,166,348),(31,309,167,349),(32,310,168,350),(33,311,141,351),(34,312,142,352),(35,313,143,353),(36,314,144,354),(37,315,145,355),(38,316,146,356),(39,317,147,357),(40,318,148,358),(41,319,149,359),(42,320,150,360),(43,321,151,361),(44,322,152,362),(45,323,153,363),(46,324,154,364),(47,325,155,337),(48,326,156,338),(49,327,157,339),(50,328,158,340),(51,329,159,341),(52,330,160,342),(53,331,161,343),(54,332,162,344),(55,333,163,345),(56,334,164,346),(57,180,376,301),(58,181,377,302),(59,182,378,303),(60,183,379,304),(61,184,380,305),(62,185,381,306),(63,186,382,307),(64,187,383,308),(65,188,384,281),(66,189,385,282),(67,190,386,283),(68,191,387,284),(69,192,388,285),(70,193,389,286),(71,194,390,287),(72,195,391,288),(73,196,392,289),(74,169,365,290),(75,170,366,291),(76,171,367,292),(77,172,368,293),(78,173,369,294),(79,174,370,295),(80,175,371,296),(81,176,372,297),(82,177,373,298),(83,178,374,299),(84,179,375,300),(85,249,118,393),(86,250,119,394),(87,251,120,395),(88,252,121,396),(89,225,122,397),(90,226,123,398),(91,227,124,399),(92,228,125,400),(93,229,126,401),(94,230,127,402),(95,231,128,403),(96,232,129,404),(97,233,130,405),(98,234,131,406),(99,235,132,407),(100,236,133,408),(101,237,134,409),(102,238,135,410),(103,239,136,411),(104,240,137,412),(105,241,138,413),(106,242,139,414),(107,243,140,415),(108,244,113,416),(109,245,114,417),(110,246,115,418),(111,247,116,419),(112,248,117,420)], [(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)], [(1,170,15,184),(2,169,16,183),(3,196,17,182),(4,195,18,181),(5,194,19,180),(6,193,20,179),(7,192,21,178),(8,191,22,177),(9,190,23,176),(10,189,24,175),(11,188,25,174),(12,187,26,173),(13,186,27,172),(14,185,28,171),(29,85,43,99),(30,112,44,98),(31,111,45,97),(32,110,46,96),(33,109,47,95),(34,108,48,94),(35,107,49,93),(36,106,50,92),(37,105,51,91),(38,104,52,90),(39,103,53,89),(40,102,54,88),(41,101,55,87),(42,100,56,86),(57,220,71,206),(58,219,72,205),(59,218,73,204),(60,217,74,203),(61,216,75,202),(62,215,76,201),(63,214,77,200),(64,213,78,199),(65,212,79,198),(66,211,80,197),(67,210,81,224),(68,209,82,223),(69,208,83,222),(70,207,84,221),(113,156,127,142),(114,155,128,141),(115,154,129,168),(116,153,130,167),(117,152,131,166),(118,151,132,165),(119,150,133,164),(120,149,134,163),(121,148,135,162),(122,147,136,161),(123,146,137,160),(124,145,138,159),(125,144,139,158),(126,143,140,157),(225,317,239,331),(226,316,240,330),(227,315,241,329),(228,314,242,328),(229,313,243,327),(230,312,244,326),(231,311,245,325),(232,310,246,324),(233,309,247,323),(234,336,248,322),(235,335,249,321),(236,334,250,320),(237,333,251,319),(238,332,252,318),(253,365,267,379),(254,392,268,378),(255,391,269,377),(256,390,270,376),(257,389,271,375),(258,388,272,374),(259,387,273,373),(260,386,274,372),(261,385,275,371),(262,384,276,370),(263,383,277,369),(264,382,278,368),(265,381,279,367),(266,380,280,366),(281,432,295,446),(282,431,296,445),(283,430,297,444),(284,429,298,443),(285,428,299,442),(286,427,300,441),(287,426,301,440),(288,425,302,439),(289,424,303,438),(290,423,304,437),(291,422,305,436),(292,421,306,435),(293,448,307,434),(294,447,308,433),(337,403,351,417),(338,402,352,416),(339,401,353,415),(340,400,354,414),(341,399,355,413),(342,398,356,412),(343,397,357,411),(344,396,358,410),(345,395,359,409),(346,394,360,408),(347,393,361,407),(348,420,362,406),(349,419,363,405),(350,418,364,404)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AF | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D7 | C4○D4 | D14 | D14 | Dic14 | C4×D7 | C4○D28 |
| kernel | C2×C4×Dic14 | C4×Dic14 | C2×C4×Dic7 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C2×C4×C28 | C22×Dic14 | C2×Dic14 | C2×C28 | C2×C42 | C2×C14 | C42 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 16 | 4 | 3 | 4 | 12 | 9 | 24 | 24 | 24 |
Matrix representation of C2×C4×Dic14 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 13 |
| 0 | 0 | 0 | 0 | 9 | 17 |
| 1 | 26 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 26 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 26 |
| 0 | 0 | 0 | 0 | 18 | 13 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,1,0,0,0,0,28,26,0,0,0,0,0,0,0,28,0,0,0,0,1,3,0,0,0,0,0,0,24,9,0,0,0,0,13,17],[1,0,0,0,0,0,26,28,0,0,0,0,0,0,21,26,0,0,0,0,21,8,0,0,0,0,0,0,16,18,0,0,0,0,26,13] >;
C2×C4×Dic14 in GAP, Magma, Sage, TeX
C_2\times C_4\times {\rm Dic}_{14} % in TeX
G:=Group("C2xC4xDic14"); // GroupNames label
G:=SmallGroup(448,920);
// by ID
G=gap.SmallGroup(448,920);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,184,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^28=1,d^2=c^14,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