G = (C2×C4).Dic14 order 448 = 26·7
9th non-split extension by C2×C4 of Dic14 acting via Dic14/C14=C22
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28).2Q8, (C2×C4).9Dic14, (C2×Dic7).10D4, (C22×C4).13D14, C22.153(D4×D7), C14.5(C22⋊Q8), C2.6(C28.3Q8), C14.1(C42.C2), C2.5(C28.6Q8), C2.8(D14.5D4), C14.17(C4.4D4), C22.87(C4○D28), (C22×C28).10C22, C22.42(C2×Dic14), C2.C42.15D7, C23.356(C22×D7), C14.18(C42⋊2C2), C22.85(D4⋊2D7), C14.C42.25C2, (C22×C14).285C23, C7⋊3(C23.83C23), C2.9(Dic7.D4), C22.41(Q8⋊2D7), C2.10(C22⋊Dic14), C14.37(C22.D4), C2.10(C23.D14), (C22×Dic7).10C22, (C2×C4⋊Dic7).9C2, (C2×C14).21(C2×Q8), C2.8(C4⋊C4⋊D7), (C2×C14).196(C2×D4), (C2×Dic7⋊C4).8C2, (C2×C14).129(C4○D4), (C7×C2.C42).11C2, SmallGroup(448,194)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4).Dic14
G = < a,b,c,d | a2=b4=c28=1, d2=b2c14, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 572 in 134 conjugacy classes, 57 normal (51 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C23, C14, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2.C42, C2.C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.83C23, Dic7⋊C4, C4⋊Dic7, C22×Dic7, C22×C28, C14.C42, C7×C2.C42, C2×Dic7⋊C4, C2×C4⋊Dic7, (C2×C4).Dic14
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, Dic14, C22×D7, C23.83C23, C2×Dic14, C4○D28, D4×D7, D4⋊2D7, Q8⋊2D7, C28.6Q8, C22⋊Dic14, C23.D14, Dic7.D4, C28.3Q8, D14.5D4, C4⋊C4⋊D7, (C2×C4).Dic14
►Regular action on 448 points
Generators in S448
(1 339)(2 340)(3 341)(4 342)(5 343)(6 344)(7 345)(8 346)(9 347)(10 348)(11 349)(12 350)(13 351)(14 352)(15 353)(16 354)(17 355)(18 356)(19 357)(20 358)(21 359)(22 360)(23 361)(24 362)(25 363)(26 364)(27 337)(28 338)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 151)(40 152)(41 153)(42 154)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 161)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 325)(58 326)(59 327)(60 328)(61 329)(62 330)(63 331)(64 332)(65 333)(66 334)(67 335)(68 336)(69 309)(70 310)(71 311)(72 312)(73 313)(74 314)(75 315)(76 316)(77 317)(78 318)(79 319)(80 320)(81 321)(82 322)(83 323)(84 324)(85 231)(86 232)(87 233)(88 234)(89 235)(90 236)(91 237)(92 238)(93 239)(94 240)(95 241)(96 242)(97 243)(98 244)(99 245)(100 246)(101 247)(102 248)(103 249)(104 250)(105 251)(106 252)(107 225)(108 226)(109 227)(110 228)(111 229)(112 230)(113 381)(114 382)(115 383)(116 384)(117 385)(118 386)(119 387)(120 388)(121 389)(122 390)(123 391)(124 392)(125 365)(126 366)(127 367)(128 368)(129 369)(130 370)(131 371)(132 372)(133 373)(134 374)(135 375)(136 376)(137 377)(138 378)(139 379)(140 380)(169 288)(170 289)(171 290)(172 291)(173 292)(174 293)(175 294)(176 295)(177 296)(178 297)(179 298)(180 299)(181 300)(182 301)(183 302)(184 303)(185 304)(186 305)(187 306)(188 307)(189 308)(190 281)(191 282)(192 283)(193 284)(194 285)(195 286)(196 287)(197 409)(198 410)(199 411)(200 412)(201 413)(202 414)(203 415)(204 416)(205 417)(206 418)(207 419)(208 420)(209 393)(210 394)(211 395)(212 396)(213 397)(214 398)(215 399)(216 400)(217 401)(218 402)(219 403)(220 404)(221 405)(222 406)(223 407)(224 408)(253 441)(254 442)(255 443)(256 444)(257 445)(258 446)(259 447)(260 448)(261 421)(262 422)(263 423)(264 424)(265 425)(266 426)(267 427)(268 428)(269 429)(270 430)(271 431)(272 432)(273 433)(274 434)(275 435)(276 436)(277 437)(278 438)(279 439)(280 440)
(1 127 308 96)(2 368 281 243)(3 129 282 98)(4 370 283 245)(5 131 284 100)(6 372 285 247)(7 133 286 102)(8 374 287 249)(9 135 288 104)(10 376 289 251)(11 137 290 106)(12 378 291 225)(13 139 292 108)(14 380 293 227)(15 113 294 110)(16 382 295 229)(17 115 296 112)(18 384 297 231)(19 117 298 86)(20 386 299 233)(21 119 300 88)(22 388 301 235)(23 121 302 90)(24 390 303 237)(25 123 304 92)(26 392 305 239)(27 125 306 94)(28 366 307 241)(29 332 197 279)(30 65 198 440)(31 334 199 253)(32 67 200 442)(33 336 201 255)(34 69 202 444)(35 310 203 257)(36 71 204 446)(37 312 205 259)(38 73 206 448)(39 314 207 261)(40 75 208 422)(41 316 209 263)(42 77 210 424)(43 318 211 265)(44 79 212 426)(45 320 213 267)(46 81 214 428)(47 322 215 269)(48 83 216 430)(49 324 217 271)(50 57 218 432)(51 326 219 273)(52 59 220 434)(53 328 221 275)(54 61 222 436)(55 330 223 277)(56 63 224 438)(58 403 433 163)(60 405 435 165)(62 407 437 167)(64 409 439 141)(66 411 441 143)(68 413 443 145)(70 415 445 147)(72 417 447 149)(74 419 421 151)(76 393 423 153)(78 395 425 155)(80 397 427 157)(82 399 429 159)(84 401 431 161)(85 356 116 178)(87 358 118 180)(89 360 120 182)(91 362 122 184)(93 364 124 186)(95 338 126 188)(97 340 128 190)(99 342 130 192)(101 344 132 194)(103 346 134 196)(105 348 136 170)(107 350 138 172)(109 352 140 174)(111 354 114 176)(142 333 410 280)(144 335 412 254)(146 309 414 256)(148 311 416 258)(150 313 418 260)(152 315 420 262)(154 317 394 264)(156 319 396 266)(158 321 398 268)(160 323 400 270)(162 325 402 272)(164 327 404 274)(166 329 406 276)(168 331 408 278)(169 250 347 375)(171 252 349 377)(173 226 351 379)(175 228 353 381)(177 230 355 383)(179 232 357 385)(181 234 359 387)(183 236 361 389)(185 238 363 391)(187 240 337 365)(189 242 339 367)(191 244 341 369)(193 246 343 371)(195 248 345 373)
(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 39 294 221)(2 206 295 52)(3 37 296 219)(4 204 297 50)(5 35 298 217)(6 202 299 48)(7 33 300 215)(8 200 301 46)(9 31 302 213)(10 198 303 44)(11 29 304 211)(12 224 305 42)(13 55 306 209)(14 222 307 40)(15 53 308 207)(16 220 281 38)(17 51 282 205)(18 218 283 36)(19 49 284 203)(20 216 285 34)(21 47 286 201)(22 214 287 32)(23 45 288 199)(24 212 289 30)(25 43 290 197)(26 210 291 56)(27 41 292 223)(28 208 293 54)(57 245 446 384)(58 369 447 230)(59 243 448 382)(60 367 421 228)(61 241 422 380)(62 365 423 226)(63 239 424 378)(64 391 425 252)(65 237 426 376)(66 389 427 250)(67 235 428 374)(68 387 429 248)(69 233 430 372)(70 385 431 246)(71 231 432 370)(72 383 433 244)(73 229 434 368)(74 381 435 242)(75 227 436 366)(76 379 437 240)(77 225 438 392)(78 377 439 238)(79 251 440 390)(80 375 441 236)(81 249 442 388)(82 373 443 234)(83 247 444 386)(84 371 445 232)(85 272 130 311)(86 324 131 257)(87 270 132 309)(88 322 133 255)(89 268 134 335)(90 320 135 253)(91 266 136 333)(92 318 137 279)(93 264 138 331)(94 316 139 277)(95 262 140 329)(96 314 113 275)(97 260 114 327)(98 312 115 273)(99 258 116 325)(100 310 117 271)(101 256 118 323)(102 336 119 269)(103 254 120 321)(104 334 121 267)(105 280 122 319)(106 332 123 265)(107 278 124 317)(108 330 125 263)(109 276 126 315)(110 328 127 261)(111 274 128 313)(112 326 129 259)(141 185 395 349)(142 362 396 170)(143 183 397 347)(144 360 398 196)(145 181 399 345)(146 358 400 194)(147 179 401 343)(148 356 402 192)(149 177 403 341)(150 354 404 190)(151 175 405 339)(152 352 406 188)(153 173 407 337)(154 350 408 186)(155 171 409 363)(156 348 410 184)(157 169 411 361)(158 346 412 182)(159 195 413 359)(160 344 414 180)(161 193 415 357)(162 342 416 178)(163 191 417 355)(164 340 418 176)(165 189 419 353)(166 338 420 174)(167 187 393 351)(168 364 394 172)
G:=sub<Sym(448)| (1,339)(2,340)(3,341)(4,342)(5,343)(6,344)(7,345)(8,346)(9,347)(10,348)(11,349)(12,350)(13,351)(14,352)(15,353)(16,354)(17,355)(18,356)(19,357)(20,358)(21,359)(22,360)(23,361)(24,362)(25,363)(26,364)(27,337)(28,338)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,325)(58,326)(59,327)(60,328)(61,329)(62,330)(63,331)(64,332)(65,333)(66,334)(67,335)(68,336)(69,309)(70,310)(71,311)(72,312)(73,313)(74,314)(75,315)(76,316)(77,317)(78,318)(79,319)(80,320)(81,321)(82,322)(83,323)(84,324)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,241)(96,242)(97,243)(98,244)(99,245)(100,246)(101,247)(102,248)(103,249)(104,250)(105,251)(106,252)(107,225)(108,226)(109,227)(110,228)(111,229)(112,230)(113,381)(114,382)(115,383)(116,384)(117,385)(118,386)(119,387)(120,388)(121,389)(122,390)(123,391)(124,392)(125,365)(126,366)(127,367)(128,368)(129,369)(130,370)(131,371)(132,372)(133,373)(134,374)(135,375)(136,376)(137,377)(138,378)(139,379)(140,380)(169,288)(170,289)(171,290)(172,291)(173,292)(174,293)(175,294)(176,295)(177,296)(178,297)(179,298)(180,299)(181,300)(182,301)(183,302)(184,303)(185,304)(186,305)(187,306)(188,307)(189,308)(190,281)(191,282)(192,283)(193,284)(194,285)(195,286)(196,287)(197,409)(198,410)(199,411)(200,412)(201,413)(202,414)(203,415)(204,416)(205,417)(206,418)(207,419)(208,420)(209,393)(210,394)(211,395)(212,396)(213,397)(214,398)(215,399)(216,400)(217,401)(218,402)(219,403)(220,404)(221,405)(222,406)(223,407)(224,408)(253,441)(254,442)(255,443)(256,444)(257,445)(258,446)(259,447)(260,448)(261,421)(262,422)(263,423)(264,424)(265,425)(266,426)(267,427)(268,428)(269,429)(270,430)(271,431)(272,432)(273,433)(274,434)(275,435)(276,436)(277,437)(278,438)(279,439)(280,440), (1,127,308,96)(2,368,281,243)(3,129,282,98)(4,370,283,245)(5,131,284,100)(6,372,285,247)(7,133,286,102)(8,374,287,249)(9,135,288,104)(10,376,289,251)(11,137,290,106)(12,378,291,225)(13,139,292,108)(14,380,293,227)(15,113,294,110)(16,382,295,229)(17,115,296,112)(18,384,297,231)(19,117,298,86)(20,386,299,233)(21,119,300,88)(22,388,301,235)(23,121,302,90)(24,390,303,237)(25,123,304,92)(26,392,305,239)(27,125,306,94)(28,366,307,241)(29,332,197,279)(30,65,198,440)(31,334,199,253)(32,67,200,442)(33,336,201,255)(34,69,202,444)(35,310,203,257)(36,71,204,446)(37,312,205,259)(38,73,206,448)(39,314,207,261)(40,75,208,422)(41,316,209,263)(42,77,210,424)(43,318,211,265)(44,79,212,426)(45,320,213,267)(46,81,214,428)(47,322,215,269)(48,83,216,430)(49,324,217,271)(50,57,218,432)(51,326,219,273)(52,59,220,434)(53,328,221,275)(54,61,222,436)(55,330,223,277)(56,63,224,438)(58,403,433,163)(60,405,435,165)(62,407,437,167)(64,409,439,141)(66,411,441,143)(68,413,443,145)(70,415,445,147)(72,417,447,149)(74,419,421,151)(76,393,423,153)(78,395,425,155)(80,397,427,157)(82,399,429,159)(84,401,431,161)(85,356,116,178)(87,358,118,180)(89,360,120,182)(91,362,122,184)(93,364,124,186)(95,338,126,188)(97,340,128,190)(99,342,130,192)(101,344,132,194)(103,346,134,196)(105,348,136,170)(107,350,138,172)(109,352,140,174)(111,354,114,176)(142,333,410,280)(144,335,412,254)(146,309,414,256)(148,311,416,258)(150,313,418,260)(152,315,420,262)(154,317,394,264)(156,319,396,266)(158,321,398,268)(160,323,400,270)(162,325,402,272)(164,327,404,274)(166,329,406,276)(168,331,408,278)(169,250,347,375)(171,252,349,377)(173,226,351,379)(175,228,353,381)(177,230,355,383)(179,232,357,385)(181,234,359,387)(183,236,361,389)(185,238,363,391)(187,240,337,365)(189,242,339,367)(191,244,341,369)(193,246,343,371)(195,248,345,373), (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), (1,39,294,221)(2,206,295,52)(3,37,296,219)(4,204,297,50)(5,35,298,217)(6,202,299,48)(7,33,300,215)(8,200,301,46)(9,31,302,213)(10,198,303,44)(11,29,304,211)(12,224,305,42)(13,55,306,209)(14,222,307,40)(15,53,308,207)(16,220,281,38)(17,51,282,205)(18,218,283,36)(19,49,284,203)(20,216,285,34)(21,47,286,201)(22,214,287,32)(23,45,288,199)(24,212,289,30)(25,43,290,197)(26,210,291,56)(27,41,292,223)(28,208,293,54)(57,245,446,384)(58,369,447,230)(59,243,448,382)(60,367,421,228)(61,241,422,380)(62,365,423,226)(63,239,424,378)(64,391,425,252)(65,237,426,376)(66,389,427,250)(67,235,428,374)(68,387,429,248)(69,233,430,372)(70,385,431,246)(71,231,432,370)(72,383,433,244)(73,229,434,368)(74,381,435,242)(75,227,436,366)(76,379,437,240)(77,225,438,392)(78,377,439,238)(79,251,440,390)(80,375,441,236)(81,249,442,388)(82,373,443,234)(83,247,444,386)(84,371,445,232)(85,272,130,311)(86,324,131,257)(87,270,132,309)(88,322,133,255)(89,268,134,335)(90,320,135,253)(91,266,136,333)(92,318,137,279)(93,264,138,331)(94,316,139,277)(95,262,140,329)(96,314,113,275)(97,260,114,327)(98,312,115,273)(99,258,116,325)(100,310,117,271)(101,256,118,323)(102,336,119,269)(103,254,120,321)(104,334,121,267)(105,280,122,319)(106,332,123,265)(107,278,124,317)(108,330,125,263)(109,276,126,315)(110,328,127,261)(111,274,128,313)(112,326,129,259)(141,185,395,349)(142,362,396,170)(143,183,397,347)(144,360,398,196)(145,181,399,345)(146,358,400,194)(147,179,401,343)(148,356,402,192)(149,177,403,341)(150,354,404,190)(151,175,405,339)(152,352,406,188)(153,173,407,337)(154,350,408,186)(155,171,409,363)(156,348,410,184)(157,169,411,361)(158,346,412,182)(159,195,413,359)(160,344,414,180)(161,193,415,357)(162,342,416,178)(163,191,417,355)(164,340,418,176)(165,189,419,353)(166,338,420,174)(167,187,393,351)(168,364,394,172)>;
G:=Group( (1,339)(2,340)(3,341)(4,342)(5,343)(6,344)(7,345)(8,346)(9,347)(10,348)(11,349)(12,350)(13,351)(14,352)(15,353)(16,354)(17,355)(18,356)(19,357)(20,358)(21,359)(22,360)(23,361)(24,362)(25,363)(26,364)(27,337)(28,338)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,325)(58,326)(59,327)(60,328)(61,329)(62,330)(63,331)(64,332)(65,333)(66,334)(67,335)(68,336)(69,309)(70,310)(71,311)(72,312)(73,313)(74,314)(75,315)(76,316)(77,317)(78,318)(79,319)(80,320)(81,321)(82,322)(83,323)(84,324)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,241)(96,242)(97,243)(98,244)(99,245)(100,246)(101,247)(102,248)(103,249)(104,250)(105,251)(106,252)(107,225)(108,226)(109,227)(110,228)(111,229)(112,230)(113,381)(114,382)(115,383)(116,384)(117,385)(118,386)(119,387)(120,388)(121,389)(122,390)(123,391)(124,392)(125,365)(126,366)(127,367)(128,368)(129,369)(130,370)(131,371)(132,372)(133,373)(134,374)(135,375)(136,376)(137,377)(138,378)(139,379)(140,380)(169,288)(170,289)(171,290)(172,291)(173,292)(174,293)(175,294)(176,295)(177,296)(178,297)(179,298)(180,299)(181,300)(182,301)(183,302)(184,303)(185,304)(186,305)(187,306)(188,307)(189,308)(190,281)(191,282)(192,283)(193,284)(194,285)(195,286)(196,287)(197,409)(198,410)(199,411)(200,412)(201,413)(202,414)(203,415)(204,416)(205,417)(206,418)(207,419)(208,420)(209,393)(210,394)(211,395)(212,396)(213,397)(214,398)(215,399)(216,400)(217,401)(218,402)(219,403)(220,404)(221,405)(222,406)(223,407)(224,408)(253,441)(254,442)(255,443)(256,444)(257,445)(258,446)(259,447)(260,448)(261,421)(262,422)(263,423)(264,424)(265,425)(266,426)(267,427)(268,428)(269,429)(270,430)(271,431)(272,432)(273,433)(274,434)(275,435)(276,436)(277,437)(278,438)(279,439)(280,440), (1,127,308,96)(2,368,281,243)(3,129,282,98)(4,370,283,245)(5,131,284,100)(6,372,285,247)(7,133,286,102)(8,374,287,249)(9,135,288,104)(10,376,289,251)(11,137,290,106)(12,378,291,225)(13,139,292,108)(14,380,293,227)(15,113,294,110)(16,382,295,229)(17,115,296,112)(18,384,297,231)(19,117,298,86)(20,386,299,233)(21,119,300,88)(22,388,301,235)(23,121,302,90)(24,390,303,237)(25,123,304,92)(26,392,305,239)(27,125,306,94)(28,366,307,241)(29,332,197,279)(30,65,198,440)(31,334,199,253)(32,67,200,442)(33,336,201,255)(34,69,202,444)(35,310,203,257)(36,71,204,446)(37,312,205,259)(38,73,206,448)(39,314,207,261)(40,75,208,422)(41,316,209,263)(42,77,210,424)(43,318,211,265)(44,79,212,426)(45,320,213,267)(46,81,214,428)(47,322,215,269)(48,83,216,430)(49,324,217,271)(50,57,218,432)(51,326,219,273)(52,59,220,434)(53,328,221,275)(54,61,222,436)(55,330,223,277)(56,63,224,438)(58,403,433,163)(60,405,435,165)(62,407,437,167)(64,409,439,141)(66,411,441,143)(68,413,443,145)(70,415,445,147)(72,417,447,149)(74,419,421,151)(76,393,423,153)(78,395,425,155)(80,397,427,157)(82,399,429,159)(84,401,431,161)(85,356,116,178)(87,358,118,180)(89,360,120,182)(91,362,122,184)(93,364,124,186)(95,338,126,188)(97,340,128,190)(99,342,130,192)(101,344,132,194)(103,346,134,196)(105,348,136,170)(107,350,138,172)(109,352,140,174)(111,354,114,176)(142,333,410,280)(144,335,412,254)(146,309,414,256)(148,311,416,258)(150,313,418,260)(152,315,420,262)(154,317,394,264)(156,319,396,266)(158,321,398,268)(160,323,400,270)(162,325,402,272)(164,327,404,274)(166,329,406,276)(168,331,408,278)(169,250,347,375)(171,252,349,377)(173,226,351,379)(175,228,353,381)(177,230,355,383)(179,232,357,385)(181,234,359,387)(183,236,361,389)(185,238,363,391)(187,240,337,365)(189,242,339,367)(191,244,341,369)(193,246,343,371)(195,248,345,373), (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), (1,39,294,221)(2,206,295,52)(3,37,296,219)(4,204,297,50)(5,35,298,217)(6,202,299,48)(7,33,300,215)(8,200,301,46)(9,31,302,213)(10,198,303,44)(11,29,304,211)(12,224,305,42)(13,55,306,209)(14,222,307,40)(15,53,308,207)(16,220,281,38)(17,51,282,205)(18,218,283,36)(19,49,284,203)(20,216,285,34)(21,47,286,201)(22,214,287,32)(23,45,288,199)(24,212,289,30)(25,43,290,197)(26,210,291,56)(27,41,292,223)(28,208,293,54)(57,245,446,384)(58,369,447,230)(59,243,448,382)(60,367,421,228)(61,241,422,380)(62,365,423,226)(63,239,424,378)(64,391,425,252)(65,237,426,376)(66,389,427,250)(67,235,428,374)(68,387,429,248)(69,233,430,372)(70,385,431,246)(71,231,432,370)(72,383,433,244)(73,229,434,368)(74,381,435,242)(75,227,436,366)(76,379,437,240)(77,225,438,392)(78,377,439,238)(79,251,440,390)(80,375,441,236)(81,249,442,388)(82,373,443,234)(83,247,444,386)(84,371,445,232)(85,272,130,311)(86,324,131,257)(87,270,132,309)(88,322,133,255)(89,268,134,335)(90,320,135,253)(91,266,136,333)(92,318,137,279)(93,264,138,331)(94,316,139,277)(95,262,140,329)(96,314,113,275)(97,260,114,327)(98,312,115,273)(99,258,116,325)(100,310,117,271)(101,256,118,323)(102,336,119,269)(103,254,120,321)(104,334,121,267)(105,280,122,319)(106,332,123,265)(107,278,124,317)(108,330,125,263)(109,276,126,315)(110,328,127,261)(111,274,128,313)(112,326,129,259)(141,185,395,349)(142,362,396,170)(143,183,397,347)(144,360,398,196)(145,181,399,345)(146,358,400,194)(147,179,401,343)(148,356,402,192)(149,177,403,341)(150,354,404,190)(151,175,405,339)(152,352,406,188)(153,173,407,337)(154,350,408,186)(155,171,409,363)(156,348,410,184)(157,169,411,361)(158,346,412,182)(159,195,413,359)(160,344,414,180)(161,193,415,357)(162,342,416,178)(163,191,417,355)(164,340,418,176)(165,189,419,353)(166,338,420,174)(167,187,393,351)(168,364,394,172) );
G=PermutationGroup([[(1,339),(2,340),(3,341),(4,342),(5,343),(6,344),(7,345),(8,346),(9,347),(10,348),(11,349),(12,350),(13,351),(14,352),(15,353),(16,354),(17,355),(18,356),(19,357),(20,358),(21,359),(22,360),(23,361),(24,362),(25,363),(26,364),(27,337),(28,338),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,148),(37,149),(38,150),(39,151),(40,152),(41,153),(42,154),(43,155),(44,156),(45,157),(46,158),(47,159),(48,160),(49,161),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,325),(58,326),(59,327),(60,328),(61,329),(62,330),(63,331),(64,332),(65,333),(66,334),(67,335),(68,336),(69,309),(70,310),(71,311),(72,312),(73,313),(74,314),(75,315),(76,316),(77,317),(78,318),(79,319),(80,320),(81,321),(82,322),(83,323),(84,324),(85,231),(86,232),(87,233),(88,234),(89,235),(90,236),(91,237),(92,238),(93,239),(94,240),(95,241),(96,242),(97,243),(98,244),(99,245),(100,246),(101,247),(102,248),(103,249),(104,250),(105,251),(106,252),(107,225),(108,226),(109,227),(110,228),(111,229),(112,230),(113,381),(114,382),(115,383),(116,384),(117,385),(118,386),(119,387),(120,388),(121,389),(122,390),(123,391),(124,392),(125,365),(126,366),(127,367),(128,368),(129,369),(130,370),(131,371),(132,372),(133,373),(134,374),(135,375),(136,376),(137,377),(138,378),(139,379),(140,380),(169,288),(170,289),(171,290),(172,291),(173,292),(174,293),(175,294),(176,295),(177,296),(178,297),(179,298),(180,299),(181,300),(182,301),(183,302),(184,303),(185,304),(186,305),(187,306),(188,307),(189,308),(190,281),(191,282),(192,283),(193,284),(194,285),(195,286),(196,287),(197,409),(198,410),(199,411),(200,412),(201,413),(202,414),(203,415),(204,416),(205,417),(206,418),(207,419),(208,420),(209,393),(210,394),(211,395),(212,396),(213,397),(214,398),(215,399),(216,400),(217,401),(218,402),(219,403),(220,404),(221,405),(222,406),(223,407),(224,408),(253,441),(254,442),(255,443),(256,444),(257,445),(258,446),(259,447),(260,448),(261,421),(262,422),(263,423),(264,424),(265,425),(266,426),(267,427),(268,428),(269,429),(270,430),(271,431),(272,432),(273,433),(274,434),(275,435),(276,436),(277,437),(278,438),(279,439),(280,440)], [(1,127,308,96),(2,368,281,243),(3,129,282,98),(4,370,283,245),(5,131,284,100),(6,372,285,247),(7,133,286,102),(8,374,287,249),(9,135,288,104),(10,376,289,251),(11,137,290,106),(12,378,291,225),(13,139,292,108),(14,380,293,227),(15,113,294,110),(16,382,295,229),(17,115,296,112),(18,384,297,231),(19,117,298,86),(20,386,299,233),(21,119,300,88),(22,388,301,235),(23,121,302,90),(24,390,303,237),(25,123,304,92),(26,392,305,239),(27,125,306,94),(28,366,307,241),(29,332,197,279),(30,65,198,440),(31,334,199,253),(32,67,200,442),(33,336,201,255),(34,69,202,444),(35,310,203,257),(36,71,204,446),(37,312,205,259),(38,73,206,448),(39,314,207,261),(40,75,208,422),(41,316,209,263),(42,77,210,424),(43,318,211,265),(44,79,212,426),(45,320,213,267),(46,81,214,428),(47,322,215,269),(48,83,216,430),(49,324,217,271),(50,57,218,432),(51,326,219,273),(52,59,220,434),(53,328,221,275),(54,61,222,436),(55,330,223,277),(56,63,224,438),(58,403,433,163),(60,405,435,165),(62,407,437,167),(64,409,439,141),(66,411,441,143),(68,413,443,145),(70,415,445,147),(72,417,447,149),(74,419,421,151),(76,393,423,153),(78,395,425,155),(80,397,427,157),(82,399,429,159),(84,401,431,161),(85,356,116,178),(87,358,118,180),(89,360,120,182),(91,362,122,184),(93,364,124,186),(95,338,126,188),(97,340,128,190),(99,342,130,192),(101,344,132,194),(103,346,134,196),(105,348,136,170),(107,350,138,172),(109,352,140,174),(111,354,114,176),(142,333,410,280),(144,335,412,254),(146,309,414,256),(148,311,416,258),(150,313,418,260),(152,315,420,262),(154,317,394,264),(156,319,396,266),(158,321,398,268),(160,323,400,270),(162,325,402,272),(164,327,404,274),(166,329,406,276),(168,331,408,278),(169,250,347,375),(171,252,349,377),(173,226,351,379),(175,228,353,381),(177,230,355,383),(179,232,357,385),(181,234,359,387),(183,236,361,389),(185,238,363,391),(187,240,337,365),(189,242,339,367),(191,244,341,369),(193,246,343,371),(195,248,345,373)], [(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)], [(1,39,294,221),(2,206,295,52),(3,37,296,219),(4,204,297,50),(5,35,298,217),(6,202,299,48),(7,33,300,215),(8,200,301,46),(9,31,302,213),(10,198,303,44),(11,29,304,211),(12,224,305,42),(13,55,306,209),(14,222,307,40),(15,53,308,207),(16,220,281,38),(17,51,282,205),(18,218,283,36),(19,49,284,203),(20,216,285,34),(21,47,286,201),(22,214,287,32),(23,45,288,199),(24,212,289,30),(25,43,290,197),(26,210,291,56),(27,41,292,223),(28,208,293,54),(57,245,446,384),(58,369,447,230),(59,243,448,382),(60,367,421,228),(61,241,422,380),(62,365,423,226),(63,239,424,378),(64,391,425,252),(65,237,426,376),(66,389,427,250),(67,235,428,374),(68,387,429,248),(69,233,430,372),(70,385,431,246),(71,231,432,370),(72,383,433,244),(73,229,434,368),(74,381,435,242),(75,227,436,366),(76,379,437,240),(77,225,438,392),(78,377,439,238),(79,251,440,390),(80,375,441,236),(81,249,442,388),(82,373,443,234),(83,247,444,386),(84,371,445,232),(85,272,130,311),(86,324,131,257),(87,270,132,309),(88,322,133,255),(89,268,134,335),(90,320,135,253),(91,266,136,333),(92,318,137,279),(93,264,138,331),(94,316,139,277),(95,262,140,329),(96,314,113,275),(97,260,114,327),(98,312,115,273),(99,258,116,325),(100,310,117,271),(101,256,118,323),(102,336,119,269),(103,254,120,321),(104,334,121,267),(105,280,122,319),(106,332,123,265),(107,278,124,317),(108,330,125,263),(109,276,126,315),(110,328,127,261),(111,274,128,313),(112,326,129,259),(141,185,395,349),(142,362,396,170),(143,183,397,347),(144,360,398,196),(145,181,399,345),(146,358,400,194),(147,179,401,343),(148,356,402,192),(149,177,403,341),(150,354,404,190),(151,175,405,339),(152,352,406,188),(153,173,407,337),(154,350,408,186),(155,171,409,363),(156,348,410,184),(157,169,411,361),(158,346,412,182),(159,195,413,359),(160,344,414,180),(161,193,415,357),(162,342,416,178),(163,191,417,355),(164,340,418,176),(165,189,419,353),(166,338,420,174),(167,187,393,351),(168,364,394,172)]])
82 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | Q8 | D7 | C4○D4 | D14 | Dic14 | C4○D28 | D4×D7 | D4⋊2D7 | Q8⋊2D7 |
| kernel | (C2×C4).Dic14 | C14.C42 | C7×C2.C42 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C2×Dic7 | C2×C28 | C2.C42 | C2×C14 | C22×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 3 | 10 | 9 | 12 | 24 | 3 | 6 | 3 |
Matrix representation of (C2×C4).Dic14 ►in GL6(𝔽29)
| 1 | 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 9 | 0 | 0 |
| 0 | 0 | 25 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 17 |
| 0 | 0 | 0 | 0 | 9 | 22 |
| 11 | 27 | 0 | 0 | 0 | 0 |
| 2 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 8 | 0 | 0 |
| 0 | 0 | 19 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 16 |
| 0 | 0 | 0 | 0 | 15 | 14 |
| 2 | 24 | 0 | 0 | 0 | 0 |
| 1 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 26 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 24 |
| 0 | 0 | 0 | 0 | 12 | 1 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,21,25,0,0,0,0,9,8,0,0,0,0,0,0,7,9,0,0,0,0,17,22],[11,2,0,0,0,0,27,5,0,0,0,0,0,0,9,19,0,0,0,0,8,20,0,0,0,0,0,0,15,15,0,0,0,0,16,14],[2,1,0,0,0,0,24,27,0,0,0,0,0,0,8,26,0,0,0,0,12,21,0,0,0,0,0,0,28,12,0,0,0,0,24,1] >;
(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(448,194);
// by ID
G=gap.SmallGroup(448,194);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,64,254,387,268,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^28=1,d^2=b^2*c^14,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations