direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.Dic14, C14⋊1(C4.Q8), C28.15(C4⋊C4), C28.62(C2×Q8), (C2×C28).15Q8, C4⋊C4.224D14, (C2×C28).131D4, C28.57(C22×C4), (C2×C4).27Dic14, (C2×C14).39SD16, C14.65(C2×SD16), C4.27(C2×Dic14), C4.15(Dic7⋊C4), (C2×C28).316C23, (C22×C14).181D4, (C22×C4).328D14, C23.96(C7⋊D4), C22.10(Q8⋊D7), C4⋊Dic7.322C22, C22.10(D4.D7), C22.24(Dic7⋊C4), (C22×C28).131C22, (C2×C7⋊C8)⋊6C4, C7⋊C8⋊24(C2×C4), C7⋊2(C2×C4.Q8), C4.85(C2×C4×D7), (C2×C4⋊C4).5D7, C2.1(C2×Q8⋊D7), (C14×C4⋊C4).4C2, C14.33(C2×C4⋊C4), (C22×C7⋊C8).4C2, C2.1(C2×D4.D7), (C2×C28).75(C2×C4), (C2×C4).150(C4×D7), C2.8(C2×Dic7⋊C4), (C2×C14).45(C4⋊C4), (C2×C14).436(C2×D4), (C2×C7⋊C8).240C22, (C2×C4⋊Dic7).30C2, C22.55(C2×C7⋊D4), (C2×C4).123(C7⋊D4), (C7×C4⋊C4).255C22, (C2×C4).416(C22×D7), SmallGroup(448,497)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4.Dic14
G = < a,b,c,d | a2=b14=c8=1, d2=b7, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c3 >
Subgroups: 452 in 130 conjugacy classes, 79 normal (27 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, Dic7, C28, C28, C28, C2×C14, C2×C14, C4.Q8, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C2×C28, C22×C14, C2×C4.Q8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C7×C4⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, C22×C28, C4.Dic14, C22×C7⋊C8, C2×C4⋊Dic7, C14×C4⋊C4, C2×C4.Dic14
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, D14, C4.Q8, C2×C4⋊C4, C2×SD16, Dic14, C4×D7, C7⋊D4, C22×D7, C2×C4.Q8, Dic7⋊C4, D4.D7, Q8⋊D7, C2×Dic14, C2×C4×D7, C2×C7⋊D4, C4.Dic14, C2×Dic7⋊C4, C2×D4.D7, C2×Q8⋊D7, C2×C4.Dic14
►Regular action on 448 points
Generators in S448
(1 198)(2 199)(3 200)(4 201)(5 202)(6 203)(7 204)(8 205)(9 206)(10 207)(11 208)(12 209)(13 210)(14 197)(15 55)(16 56)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 99)(39 100)(40 101)(41 102)(42 103)(57 398)(58 399)(59 400)(60 401)(61 402)(62 403)(63 404)(64 405)(65 406)(66 393)(67 394)(68 395)(69 396)(70 397)(71 328)(72 329)(73 330)(74 331)(75 332)(76 333)(77 334)(78 335)(79 336)(80 323)(81 324)(82 325)(83 326)(84 327)(85 432)(86 433)(87 434)(88 421)(89 422)(90 423)(91 424)(92 425)(93 426)(94 427)(95 428)(96 429)(97 430)(98 431)(113 322)(114 309)(115 310)(116 311)(117 312)(118 313)(119 314)(120 315)(121 316)(122 317)(123 318)(124 319)(125 320)(126 321)(127 295)(128 296)(129 297)(130 298)(131 299)(132 300)(133 301)(134 302)(135 303)(136 304)(137 305)(138 306)(139 307)(140 308)(141 351)(142 352)(143 353)(144 354)(145 355)(146 356)(147 357)(148 358)(149 359)(150 360)(151 361)(152 362)(153 363)(154 364)(155 234)(156 235)(157 236)(158 237)(159 238)(160 225)(161 226)(162 227)(163 228)(164 229)(165 230)(166 231)(167 232)(168 233)(169 407)(170 408)(171 409)(172 410)(173 411)(174 412)(175 413)(176 414)(177 415)(178 416)(179 417)(180 418)(181 419)(182 420)(183 387)(184 388)(185 389)(186 390)(187 391)(188 392)(189 379)(190 380)(191 381)(192 382)(193 383)(194 384)(195 385)(196 386)(211 271)(212 272)(213 273)(214 274)(215 275)(216 276)(217 277)(218 278)(219 279)(220 280)(221 267)(222 268)(223 269)(224 270)(239 345)(240 346)(241 347)(242 348)(243 349)(244 350)(245 337)(246 338)(247 339)(248 340)(249 341)(250 342)(251 343)(252 344)(253 441)(254 442)(255 443)(256 444)(257 445)(258 446)(259 447)(260 448)(261 435)(262 436)(263 437)(264 438)(265 439)(266 440)(281 365)(282 366)(283 367)(284 368)(285 369)(286 370)(287 371)(288 372)(289 373)(290 374)(291 375)(292 376)(293 377)(294 378)
(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)
(1 379 128 107 443 49 286 267)(2 392 129 106 444 48 287 280)(3 391 130 105 445 47 288 279)(4 390 131 104 446 46 289 278)(5 389 132 103 447 45 290 277)(6 388 133 102 448 44 291 276)(7 387 134 101 435 43 292 275)(8 386 135 100 436 56 293 274)(9 385 136 99 437 55 294 273)(10 384 137 112 438 54 281 272)(11 383 138 111 439 53 282 271)(12 382 139 110 440 52 283 270)(13 381 140 109 441 51 284 269)(14 380 127 108 442 50 285 268)(15 378 213 206 195 304 38 263)(16 377 214 205 196 303 39 262)(17 376 215 204 183 302 40 261)(18 375 216 203 184 301 41 260)(19 374 217 202 185 300 42 259)(20 373 218 201 186 299 29 258)(21 372 219 200 187 298 30 257)(22 371 220 199 188 297 31 256)(23 370 221 198 189 296 32 255)(24 369 222 197 190 295 33 254)(25 368 223 210 191 308 34 253)(26 367 224 209 192 307 35 266)(27 366 211 208 193 306 36 265)(28 365 212 207 194 305 37 264)(57 85 114 229 145 78 244 410)(58 98 115 228 146 77 245 409)(59 97 116 227 147 76 246 408)(60 96 117 226 148 75 247 407)(61 95 118 225 149 74 248 420)(62 94 119 238 150 73 249 419)(63 93 120 237 151 72 250 418)(64 92 121 236 152 71 251 417)(65 91 122 235 153 84 252 416)(66 90 123 234 154 83 239 415)(67 89 124 233 141 82 240 414)(68 88 125 232 142 81 241 413)(69 87 126 231 143 80 242 412)(70 86 113 230 144 79 243 411)(155 364 326 345 177 393 423 318)(156 363 327 344 178 406 424 317)(157 362 328 343 179 405 425 316)(158 361 329 342 180 404 426 315)(159 360 330 341 181 403 427 314)(160 359 331 340 182 402 428 313)(161 358 332 339 169 401 429 312)(162 357 333 338 170 400 430 311)(163 356 334 337 171 399 431 310)(164 355 335 350 172 398 432 309)(165 354 336 349 173 397 433 322)(166 353 323 348 174 396 434 321)(167 352 324 347 175 395 421 320)(168 351 325 346 176 394 422 319)
(1 434 8 427)(2 421 9 428)(3 422 10 429)(4 423 11 430)(5 424 12 431)(6 425 13 432)(7 426 14 433)(15 61 22 68)(16 62 23 69)(17 63 24 70)(18 64 25 57)(19 65 26 58)(20 66 27 59)(21 67 28 60)(29 123 36 116)(30 124 37 117)(31 125 38 118)(32 126 39 119)(33 113 40 120)(34 114 41 121)(35 115 42 122)(43 404 50 397)(44 405 51 398)(45 406 52 399)(46 393 53 400)(47 394 54 401)(48 395 55 402)(49 396 56 403)(71 253 78 260)(72 254 79 261)(73 255 80 262)(74 256 81 263)(75 257 82 264)(76 258 83 265)(77 259 84 266)(85 203 92 210)(86 204 93 197)(87 205 94 198)(88 206 95 199)(89 207 96 200)(90 208 97 201)(91 209 98 202)(99 313 106 320)(100 314 107 321)(101 315 108 322)(102 316 109 309)(103 317 110 310)(104 318 111 311)(105 319 112 312)(127 173 134 180)(128 174 135 181)(129 175 136 182)(130 176 137 169)(131 177 138 170)(132 178 139 171)(133 179 140 172)(141 194 148 187)(142 195 149 188)(143 196 150 189)(144 183 151 190)(145 184 152 191)(146 185 153 192)(147 186 154 193)(155 282 162 289)(156 283 163 290)(157 284 164 291)(158 285 165 292)(159 286 166 293)(160 287 167 294)(161 288 168 281)(211 246 218 239)(212 247 219 240)(213 248 220 241)(214 249 221 242)(215 250 222 243)(216 251 223 244)(217 252 224 245)(225 371 232 378)(226 372 233 365)(227 373 234 366)(228 374 235 367)(229 375 236 368)(230 376 237 369)(231 377 238 370)(267 348 274 341)(268 349 275 342)(269 350 276 343)(270 337 277 344)(271 338 278 345)(272 339 279 346)(273 340 280 347)(295 411 302 418)(296 412 303 419)(297 413 304 420)(298 414 305 407)(299 415 306 408)(300 416 307 409)(301 417 308 410)(323 436 330 443)(324 437 331 444)(325 438 332 445)(326 439 333 446)(327 440 334 447)(328 441 335 448)(329 442 336 435)(351 384 358 391)(352 385 359 392)(353 386 360 379)(354 387 361 380)(355 388 362 381)(356 389 363 382)(357 390 364 383)
G:=sub<Sym(448)| (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,197)(15,55)(16,56)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,99)(39,100)(40,101)(41,102)(42,103)(57,398)(58,399)(59,400)(60,401)(61,402)(62,403)(63,404)(64,405)(65,406)(66,393)(67,394)(68,395)(69,396)(70,397)(71,328)(72,329)(73,330)(74,331)(75,332)(76,333)(77,334)(78,335)(79,336)(80,323)(81,324)(82,325)(83,326)(84,327)(85,432)(86,433)(87,434)(88,421)(89,422)(90,423)(91,424)(92,425)(93,426)(94,427)(95,428)(96,429)(97,430)(98,431)(113,322)(114,309)(115,310)(116,311)(117,312)(118,313)(119,314)(120,315)(121,316)(122,317)(123,318)(124,319)(125,320)(126,321)(127,295)(128,296)(129,297)(130,298)(131,299)(132,300)(133,301)(134,302)(135,303)(136,304)(137,305)(138,306)(139,307)(140,308)(141,351)(142,352)(143,353)(144,354)(145,355)(146,356)(147,357)(148,358)(149,359)(150,360)(151,361)(152,362)(153,363)(154,364)(155,234)(156,235)(157,236)(158,237)(159,238)(160,225)(161,226)(162,227)(163,228)(164,229)(165,230)(166,231)(167,232)(168,233)(169,407)(170,408)(171,409)(172,410)(173,411)(174,412)(175,413)(176,414)(177,415)(178,416)(179,417)(180,418)(181,419)(182,420)(183,387)(184,388)(185,389)(186,390)(187,391)(188,392)(189,379)(190,380)(191,381)(192,382)(193,383)(194,384)(195,385)(196,386)(211,271)(212,272)(213,273)(214,274)(215,275)(216,276)(217,277)(218,278)(219,279)(220,280)(221,267)(222,268)(223,269)(224,270)(239,345)(240,346)(241,347)(242,348)(243,349)(244,350)(245,337)(246,338)(247,339)(248,340)(249,341)(250,342)(251,343)(252,344)(253,441)(254,442)(255,443)(256,444)(257,445)(258,446)(259,447)(260,448)(261,435)(262,436)(263,437)(264,438)(265,439)(266,440)(281,365)(282,366)(283,367)(284,368)(285,369)(286,370)(287,371)(288,372)(289,373)(290,374)(291,375)(292,376)(293,377)(294,378), (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), (1,379,128,107,443,49,286,267)(2,392,129,106,444,48,287,280)(3,391,130,105,445,47,288,279)(4,390,131,104,446,46,289,278)(5,389,132,103,447,45,290,277)(6,388,133,102,448,44,291,276)(7,387,134,101,435,43,292,275)(8,386,135,100,436,56,293,274)(9,385,136,99,437,55,294,273)(10,384,137,112,438,54,281,272)(11,383,138,111,439,53,282,271)(12,382,139,110,440,52,283,270)(13,381,140,109,441,51,284,269)(14,380,127,108,442,50,285,268)(15,378,213,206,195,304,38,263)(16,377,214,205,196,303,39,262)(17,376,215,204,183,302,40,261)(18,375,216,203,184,301,41,260)(19,374,217,202,185,300,42,259)(20,373,218,201,186,299,29,258)(21,372,219,200,187,298,30,257)(22,371,220,199,188,297,31,256)(23,370,221,198,189,296,32,255)(24,369,222,197,190,295,33,254)(25,368,223,210,191,308,34,253)(26,367,224,209,192,307,35,266)(27,366,211,208,193,306,36,265)(28,365,212,207,194,305,37,264)(57,85,114,229,145,78,244,410)(58,98,115,228,146,77,245,409)(59,97,116,227,147,76,246,408)(60,96,117,226,148,75,247,407)(61,95,118,225,149,74,248,420)(62,94,119,238,150,73,249,419)(63,93,120,237,151,72,250,418)(64,92,121,236,152,71,251,417)(65,91,122,235,153,84,252,416)(66,90,123,234,154,83,239,415)(67,89,124,233,141,82,240,414)(68,88,125,232,142,81,241,413)(69,87,126,231,143,80,242,412)(70,86,113,230,144,79,243,411)(155,364,326,345,177,393,423,318)(156,363,327,344,178,406,424,317)(157,362,328,343,179,405,425,316)(158,361,329,342,180,404,426,315)(159,360,330,341,181,403,427,314)(160,359,331,340,182,402,428,313)(161,358,332,339,169,401,429,312)(162,357,333,338,170,400,430,311)(163,356,334,337,171,399,431,310)(164,355,335,350,172,398,432,309)(165,354,336,349,173,397,433,322)(166,353,323,348,174,396,434,321)(167,352,324,347,175,395,421,320)(168,351,325,346,176,394,422,319), (1,434,8,427)(2,421,9,428)(3,422,10,429)(4,423,11,430)(5,424,12,431)(6,425,13,432)(7,426,14,433)(15,61,22,68)(16,62,23,69)(17,63,24,70)(18,64,25,57)(19,65,26,58)(20,66,27,59)(21,67,28,60)(29,123,36,116)(30,124,37,117)(31,125,38,118)(32,126,39,119)(33,113,40,120)(34,114,41,121)(35,115,42,122)(43,404,50,397)(44,405,51,398)(45,406,52,399)(46,393,53,400)(47,394,54,401)(48,395,55,402)(49,396,56,403)(71,253,78,260)(72,254,79,261)(73,255,80,262)(74,256,81,263)(75,257,82,264)(76,258,83,265)(77,259,84,266)(85,203,92,210)(86,204,93,197)(87,205,94,198)(88,206,95,199)(89,207,96,200)(90,208,97,201)(91,209,98,202)(99,313,106,320)(100,314,107,321)(101,315,108,322)(102,316,109,309)(103,317,110,310)(104,318,111,311)(105,319,112,312)(127,173,134,180)(128,174,135,181)(129,175,136,182)(130,176,137,169)(131,177,138,170)(132,178,139,171)(133,179,140,172)(141,194,148,187)(142,195,149,188)(143,196,150,189)(144,183,151,190)(145,184,152,191)(146,185,153,192)(147,186,154,193)(155,282,162,289)(156,283,163,290)(157,284,164,291)(158,285,165,292)(159,286,166,293)(160,287,167,294)(161,288,168,281)(211,246,218,239)(212,247,219,240)(213,248,220,241)(214,249,221,242)(215,250,222,243)(216,251,223,244)(217,252,224,245)(225,371,232,378)(226,372,233,365)(227,373,234,366)(228,374,235,367)(229,375,236,368)(230,376,237,369)(231,377,238,370)(267,348,274,341)(268,349,275,342)(269,350,276,343)(270,337,277,344)(271,338,278,345)(272,339,279,346)(273,340,280,347)(295,411,302,418)(296,412,303,419)(297,413,304,420)(298,414,305,407)(299,415,306,408)(300,416,307,409)(301,417,308,410)(323,436,330,443)(324,437,331,444)(325,438,332,445)(326,439,333,446)(327,440,334,447)(328,441,335,448)(329,442,336,435)(351,384,358,391)(352,385,359,392)(353,386,360,379)(354,387,361,380)(355,388,362,381)(356,389,363,382)(357,390,364,383)>;
G:=Group( (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,197)(15,55)(16,56)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,99)(39,100)(40,101)(41,102)(42,103)(57,398)(58,399)(59,400)(60,401)(61,402)(62,403)(63,404)(64,405)(65,406)(66,393)(67,394)(68,395)(69,396)(70,397)(71,328)(72,329)(73,330)(74,331)(75,332)(76,333)(77,334)(78,335)(79,336)(80,323)(81,324)(82,325)(83,326)(84,327)(85,432)(86,433)(87,434)(88,421)(89,422)(90,423)(91,424)(92,425)(93,426)(94,427)(95,428)(96,429)(97,430)(98,431)(113,322)(114,309)(115,310)(116,311)(117,312)(118,313)(119,314)(120,315)(121,316)(122,317)(123,318)(124,319)(125,320)(126,321)(127,295)(128,296)(129,297)(130,298)(131,299)(132,300)(133,301)(134,302)(135,303)(136,304)(137,305)(138,306)(139,307)(140,308)(141,351)(142,352)(143,353)(144,354)(145,355)(146,356)(147,357)(148,358)(149,359)(150,360)(151,361)(152,362)(153,363)(154,364)(155,234)(156,235)(157,236)(158,237)(159,238)(160,225)(161,226)(162,227)(163,228)(164,229)(165,230)(166,231)(167,232)(168,233)(169,407)(170,408)(171,409)(172,410)(173,411)(174,412)(175,413)(176,414)(177,415)(178,416)(179,417)(180,418)(181,419)(182,420)(183,387)(184,388)(185,389)(186,390)(187,391)(188,392)(189,379)(190,380)(191,381)(192,382)(193,383)(194,384)(195,385)(196,386)(211,271)(212,272)(213,273)(214,274)(215,275)(216,276)(217,277)(218,278)(219,279)(220,280)(221,267)(222,268)(223,269)(224,270)(239,345)(240,346)(241,347)(242,348)(243,349)(244,350)(245,337)(246,338)(247,339)(248,340)(249,341)(250,342)(251,343)(252,344)(253,441)(254,442)(255,443)(256,444)(257,445)(258,446)(259,447)(260,448)(261,435)(262,436)(263,437)(264,438)(265,439)(266,440)(281,365)(282,366)(283,367)(284,368)(285,369)(286,370)(287,371)(288,372)(289,373)(290,374)(291,375)(292,376)(293,377)(294,378), (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), (1,379,128,107,443,49,286,267)(2,392,129,106,444,48,287,280)(3,391,130,105,445,47,288,279)(4,390,131,104,446,46,289,278)(5,389,132,103,447,45,290,277)(6,388,133,102,448,44,291,276)(7,387,134,101,435,43,292,275)(8,386,135,100,436,56,293,274)(9,385,136,99,437,55,294,273)(10,384,137,112,438,54,281,272)(11,383,138,111,439,53,282,271)(12,382,139,110,440,52,283,270)(13,381,140,109,441,51,284,269)(14,380,127,108,442,50,285,268)(15,378,213,206,195,304,38,263)(16,377,214,205,196,303,39,262)(17,376,215,204,183,302,40,261)(18,375,216,203,184,301,41,260)(19,374,217,202,185,300,42,259)(20,373,218,201,186,299,29,258)(21,372,219,200,187,298,30,257)(22,371,220,199,188,297,31,256)(23,370,221,198,189,296,32,255)(24,369,222,197,190,295,33,254)(25,368,223,210,191,308,34,253)(26,367,224,209,192,307,35,266)(27,366,211,208,193,306,36,265)(28,365,212,207,194,305,37,264)(57,85,114,229,145,78,244,410)(58,98,115,228,146,77,245,409)(59,97,116,227,147,76,246,408)(60,96,117,226,148,75,247,407)(61,95,118,225,149,74,248,420)(62,94,119,238,150,73,249,419)(63,93,120,237,151,72,250,418)(64,92,121,236,152,71,251,417)(65,91,122,235,153,84,252,416)(66,90,123,234,154,83,239,415)(67,89,124,233,141,82,240,414)(68,88,125,232,142,81,241,413)(69,87,126,231,143,80,242,412)(70,86,113,230,144,79,243,411)(155,364,326,345,177,393,423,318)(156,363,327,344,178,406,424,317)(157,362,328,343,179,405,425,316)(158,361,329,342,180,404,426,315)(159,360,330,341,181,403,427,314)(160,359,331,340,182,402,428,313)(161,358,332,339,169,401,429,312)(162,357,333,338,170,400,430,311)(163,356,334,337,171,399,431,310)(164,355,335,350,172,398,432,309)(165,354,336,349,173,397,433,322)(166,353,323,348,174,396,434,321)(167,352,324,347,175,395,421,320)(168,351,325,346,176,394,422,319), (1,434,8,427)(2,421,9,428)(3,422,10,429)(4,423,11,430)(5,424,12,431)(6,425,13,432)(7,426,14,433)(15,61,22,68)(16,62,23,69)(17,63,24,70)(18,64,25,57)(19,65,26,58)(20,66,27,59)(21,67,28,60)(29,123,36,116)(30,124,37,117)(31,125,38,118)(32,126,39,119)(33,113,40,120)(34,114,41,121)(35,115,42,122)(43,404,50,397)(44,405,51,398)(45,406,52,399)(46,393,53,400)(47,394,54,401)(48,395,55,402)(49,396,56,403)(71,253,78,260)(72,254,79,261)(73,255,80,262)(74,256,81,263)(75,257,82,264)(76,258,83,265)(77,259,84,266)(85,203,92,210)(86,204,93,197)(87,205,94,198)(88,206,95,199)(89,207,96,200)(90,208,97,201)(91,209,98,202)(99,313,106,320)(100,314,107,321)(101,315,108,322)(102,316,109,309)(103,317,110,310)(104,318,111,311)(105,319,112,312)(127,173,134,180)(128,174,135,181)(129,175,136,182)(130,176,137,169)(131,177,138,170)(132,178,139,171)(133,179,140,172)(141,194,148,187)(142,195,149,188)(143,196,150,189)(144,183,151,190)(145,184,152,191)(146,185,153,192)(147,186,154,193)(155,282,162,289)(156,283,163,290)(157,284,164,291)(158,285,165,292)(159,286,166,293)(160,287,167,294)(161,288,168,281)(211,246,218,239)(212,247,219,240)(213,248,220,241)(214,249,221,242)(215,250,222,243)(216,251,223,244)(217,252,224,245)(225,371,232,378)(226,372,233,365)(227,373,234,366)(228,374,235,367)(229,375,236,368)(230,376,237,369)(231,377,238,370)(267,348,274,341)(268,349,275,342)(269,350,276,343)(270,337,277,344)(271,338,278,345)(272,339,279,346)(273,340,280,347)(295,411,302,418)(296,412,303,419)(297,413,304,420)(298,414,305,407)(299,415,306,408)(300,416,307,409)(301,417,308,410)(323,436,330,443)(324,437,331,444)(325,438,332,445)(326,439,333,446)(327,440,334,447)(328,441,335,448)(329,442,336,435)(351,384,358,391)(352,385,359,392)(353,386,360,379)(354,387,361,380)(355,388,362,381)(356,389,363,382)(357,390,364,383) );
G=PermutationGroup([[(1,198),(2,199),(3,200),(4,201),(5,202),(6,203),(7,204),(8,205),(9,206),(10,207),(11,208),(12,209),(13,210),(14,197),(15,55),(16,56),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,99),(39,100),(40,101),(41,102),(42,103),(57,398),(58,399),(59,400),(60,401),(61,402),(62,403),(63,404),(64,405),(65,406),(66,393),(67,394),(68,395),(69,396),(70,397),(71,328),(72,329),(73,330),(74,331),(75,332),(76,333),(77,334),(78,335),(79,336),(80,323),(81,324),(82,325),(83,326),(84,327),(85,432),(86,433),(87,434),(88,421),(89,422),(90,423),(91,424),(92,425),(93,426),(94,427),(95,428),(96,429),(97,430),(98,431),(113,322),(114,309),(115,310),(116,311),(117,312),(118,313),(119,314),(120,315),(121,316),(122,317),(123,318),(124,319),(125,320),(126,321),(127,295),(128,296),(129,297),(130,298),(131,299),(132,300),(133,301),(134,302),(135,303),(136,304),(137,305),(138,306),(139,307),(140,308),(141,351),(142,352),(143,353),(144,354),(145,355),(146,356),(147,357),(148,358),(149,359),(150,360),(151,361),(152,362),(153,363),(154,364),(155,234),(156,235),(157,236),(158,237),(159,238),(160,225),(161,226),(162,227),(163,228),(164,229),(165,230),(166,231),(167,232),(168,233),(169,407),(170,408),(171,409),(172,410),(173,411),(174,412),(175,413),(176,414),(177,415),(178,416),(179,417),(180,418),(181,419),(182,420),(183,387),(184,388),(185,389),(186,390),(187,391),(188,392),(189,379),(190,380),(191,381),(192,382),(193,383),(194,384),(195,385),(196,386),(211,271),(212,272),(213,273),(214,274),(215,275),(216,276),(217,277),(218,278),(219,279),(220,280),(221,267),(222,268),(223,269),(224,270),(239,345),(240,346),(241,347),(242,348),(243,349),(244,350),(245,337),(246,338),(247,339),(248,340),(249,341),(250,342),(251,343),(252,344),(253,441),(254,442),(255,443),(256,444),(257,445),(258,446),(259,447),(260,448),(261,435),(262,436),(263,437),(264,438),(265,439),(266,440),(281,365),(282,366),(283,367),(284,368),(285,369),(286,370),(287,371),(288,372),(289,373),(290,374),(291,375),(292,376),(293,377),(294,378)], [(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)], [(1,379,128,107,443,49,286,267),(2,392,129,106,444,48,287,280),(3,391,130,105,445,47,288,279),(4,390,131,104,446,46,289,278),(5,389,132,103,447,45,290,277),(6,388,133,102,448,44,291,276),(7,387,134,101,435,43,292,275),(8,386,135,100,436,56,293,274),(9,385,136,99,437,55,294,273),(10,384,137,112,438,54,281,272),(11,383,138,111,439,53,282,271),(12,382,139,110,440,52,283,270),(13,381,140,109,441,51,284,269),(14,380,127,108,442,50,285,268),(15,378,213,206,195,304,38,263),(16,377,214,205,196,303,39,262),(17,376,215,204,183,302,40,261),(18,375,216,203,184,301,41,260),(19,374,217,202,185,300,42,259),(20,373,218,201,186,299,29,258),(21,372,219,200,187,298,30,257),(22,371,220,199,188,297,31,256),(23,370,221,198,189,296,32,255),(24,369,222,197,190,295,33,254),(25,368,223,210,191,308,34,253),(26,367,224,209,192,307,35,266),(27,366,211,208,193,306,36,265),(28,365,212,207,194,305,37,264),(57,85,114,229,145,78,244,410),(58,98,115,228,146,77,245,409),(59,97,116,227,147,76,246,408),(60,96,117,226,148,75,247,407),(61,95,118,225,149,74,248,420),(62,94,119,238,150,73,249,419),(63,93,120,237,151,72,250,418),(64,92,121,236,152,71,251,417),(65,91,122,235,153,84,252,416),(66,90,123,234,154,83,239,415),(67,89,124,233,141,82,240,414),(68,88,125,232,142,81,241,413),(69,87,126,231,143,80,242,412),(70,86,113,230,144,79,243,411),(155,364,326,345,177,393,423,318),(156,363,327,344,178,406,424,317),(157,362,328,343,179,405,425,316),(158,361,329,342,180,404,426,315),(159,360,330,341,181,403,427,314),(160,359,331,340,182,402,428,313),(161,358,332,339,169,401,429,312),(162,357,333,338,170,400,430,311),(163,356,334,337,171,399,431,310),(164,355,335,350,172,398,432,309),(165,354,336,349,173,397,433,322),(166,353,323,348,174,396,434,321),(167,352,324,347,175,395,421,320),(168,351,325,346,176,394,422,319)], [(1,434,8,427),(2,421,9,428),(3,422,10,429),(4,423,11,430),(5,424,12,431),(6,425,13,432),(7,426,14,433),(15,61,22,68),(16,62,23,69),(17,63,24,70),(18,64,25,57),(19,65,26,58),(20,66,27,59),(21,67,28,60),(29,123,36,116),(30,124,37,117),(31,125,38,118),(32,126,39,119),(33,113,40,120),(34,114,41,121),(35,115,42,122),(43,404,50,397),(44,405,51,398),(45,406,52,399),(46,393,53,400),(47,394,54,401),(48,395,55,402),(49,396,56,403),(71,253,78,260),(72,254,79,261),(73,255,80,262),(74,256,81,263),(75,257,82,264),(76,258,83,265),(77,259,84,266),(85,203,92,210),(86,204,93,197),(87,205,94,198),(88,206,95,199),(89,207,96,200),(90,208,97,201),(91,209,98,202),(99,313,106,320),(100,314,107,321),(101,315,108,322),(102,316,109,309),(103,317,110,310),(104,318,111,311),(105,319,112,312),(127,173,134,180),(128,174,135,181),(129,175,136,182),(130,176,137,169),(131,177,138,170),(132,178,139,171),(133,179,140,172),(141,194,148,187),(142,195,149,188),(143,196,150,189),(144,183,151,190),(145,184,152,191),(146,185,153,192),(147,186,154,193),(155,282,162,289),(156,283,163,290),(157,284,164,291),(158,285,165,292),(159,286,166,293),(160,287,167,294),(161,288,168,281),(211,246,218,239),(212,247,219,240),(213,248,220,241),(214,249,221,242),(215,250,222,243),(216,251,223,244),(217,252,224,245),(225,371,232,378),(226,372,233,365),(227,373,234,366),(228,374,235,367),(229,375,236,368),(230,376,237,369),(231,377,238,370),(267,348,274,341),(268,349,275,342),(269,350,276,343),(270,337,277,344),(271,338,278,345),(272,339,279,346),(273,340,280,347),(295,411,302,418),(296,412,303,419),(297,413,304,420),(298,414,305,407),(299,415,306,408),(300,416,307,409),(301,417,308,410),(323,436,330,443),(324,437,331,444),(325,438,332,445),(326,439,333,446),(327,440,334,447),(328,441,335,448),(329,442,336,435),(351,384,358,391),(352,385,359,392),(353,386,360,379),(354,387,361,380),(355,388,362,381),(356,389,363,382),(357,390,364,383)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D7 | SD16 | D14 | D14 | Dic14 | C4×D7 | C7⋊D4 | C7⋊D4 | D4.D7 | Q8⋊D7 |
| kernel | C2×C4.Dic14 | C4.Dic14 | C22×C7⋊C8 | C2×C4⋊Dic7 | C14×C4⋊C4 | C2×C7⋊C8 | C2×C28 | C2×C28 | C22×C14 | C2×C4⋊C4 | C2×C14 | C4⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 3 | 8 | 6 | 3 | 12 | 12 | 6 | 6 | 6 | 6 |
Matrix representation of C2×C4.Dic14 ►in GL5(𝔽113)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 104 | 88 |
| 0 | 0 | 0 | 104 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 13 | 100 | 0 | 0 |
| 0 | 13 | 13 | 0 | 0 |
| 0 | 0 | 0 | 58 | 93 |
| 0 | 0 | 0 | 27 | 55 |
| 98 | 0 | 0 | 0 | 0 |
| 0 | 14 | 101 | 0 | 0 |
| 0 | 101 | 99 | 0 | 0 |
| 0 | 0 | 0 | 58 | 46 |
| 0 | 0 | 0 | 3 | 55 |
G:=sub<GL(5,GF(113))| [1,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112],[112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,104,104,0,0,0,88,0],[1,0,0,0,0,0,13,13,0,0,0,100,13,0,0,0,0,0,58,27,0,0,0,93,55],[98,0,0,0,0,0,14,101,0,0,0,101,99,0,0,0,0,0,58,3,0,0,0,46,55] >;
C2×C4.Dic14 in GAP, Magma, Sage, TeX
C_2\times C_4.{\rm Dic}_{14} % in TeX
G:=Group("C2xC4.Dic14"); // GroupNames label
G:=SmallGroup(448,497);
// by ID
G=gap.SmallGroup(448,497);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,1094,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^14=c^8=1,d^2=b^7,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations