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G = (C2×C4)⋊Dic14  order 448 = 26·7

2nd semidirect product of C2×C4 and Dic14 acting via Dic14/C14=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C28)⋊7Q8, (C2×Dic7)⋊2Q8, (C2×C4)⋊2Dic14, (C2×C28).52D4, C2.14(C28⋊Q8), C14.21(C4⋊Q8), C14.40C22≀C2, C22.45(Q8×D7), (C2×Dic7).58D4, C22.244(D4×D7), (C22×C4).98D14, C2.9(C23⋊D14), C14.58(C22⋊Q8), C2.20(D14⋊Q8), C2.5(Dic7⋊Q8), C2.9(C28.48D4), (C22×C28).63C22, (C22×Dic14).5C2, C22.47(C2×Dic14), C23.374(C22×D7), C22.102(C4○D28), C14.C42.37C2, (C22×C14).343C23, C7⋊2(C23.78C23), (C22×Dic7).52C22, (C2×C4⋊C4).16D7, (C14×C4⋊C4).21C2, (C2×C14).78(C2×Q8), (C2×C14).330(C2×D4), (C2×C4).35(C7⋊D4), (C2×C14).82(C4○D4), (C2×Dic7⋊C4).13C2, C22.134(C2×C7⋊D4), SmallGroup(448,513)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C14 — (C2×C4)⋊Dic14
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C22×Dic14 — (C2×C4)⋊Dic14
Lower central
C7 — C22×C14 — (C2×C4)⋊Dic14
Upper central
C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C4)⋊Dic14
 G = < a,b,c,d | a2=b4=c28=1, d2=c14, dbd-1=ab=ba, ac=ca, ad=da, cbc-1=b-1, dcd-1=c-1 >

Subgroups: 804 in 182 conjugacy classes, 67 normal (27 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C22×Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.78C23, Dic7⋊C4, C7×C4⋊C4, C2×Dic14, C22×Dic7, C22×C28, C14.C42, C14.C42, C2×Dic7⋊C4, C14×C4⋊C4, C22×Dic14, (C2×C4)⋊Dic14
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22≀C2, C22⋊Q8, C4⋊Q8, Dic14, C7⋊D4, C22×D7, C23.78C23, C2×Dic14, C4○D28, D4×D7, Q8×D7, C2×C7⋊D4, C28⋊Q8, D14⋊Q8, C28.48D4, C23⋊D14, Dic7⋊Q8, (C2×C4)⋊Dic14

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

(1 42 448 73)(2 74 421 43)(3 44 422 75)(4 76 423 45)(5 46 424 77)(6 78 425 47)(7 48 426 79)(8 80 427 49)(9 50 428 81)(10 82 429 51)(11 52 430 83)(12 84 431 53)(13 54 432 57)(14 58 433 55)(15 56 434 59)(16 60 435 29)(17 30 436 61)(18 62 437 31)(19 32 438 63)(20 64 439 33)(21 34 440 65)(22 66 441 35)(23 36 442 67)(24 68 443 37)(25 38 444 69)(26 70 445 39)(27 40 446 71)(28 72 447 41)(85 284 192 167)(86 168 193 285)(87 286 194 141)(88 142 195 287)(89 288 196 143)(90 144 169 289)(91 290 170 145)(92 146 171 291)(93 292 172 147)(94 148 173 293)(95 294 174 149)(96 150 175 295)(97 296 176 151)(98 152 177 297)(99 298 178 153)(100 154 179 299)(101 300 180 155)(102 156 181 301)(103 302 182 157)(104 158 183 303)(105 304 184 159)(106 160 185 305)(107 306 186 161)(108 162 187 307)(109 308 188 163)(110 164 189 281)(111 282 190 165)(112 166 191 283)(113 335 408 246)(114 247 409 336)(115 309 410 248)(116 249 411 310)(117 311 412 250)(118 251 413 312)(119 313 414 252)(120 225 415 314)(121 315 416 226)(122 227 417 316)(123 317 418 228)(124 229 419 318)(125 319 420 230)(126 231 393 320)(127 321 394 232)(128 233 395 322)(129 323 396 234)(130 235 397 324)(131 325 398 236)(132 237 399 326)(133 327 400 238)(134 239 401 328)(135 329 402 240)(136 241 403 330)(137 331 404 242)(138 243 405 332)(139 333 406 244)(140 245 407 334)(197 383 253 352)(198 353 254 384)(199 385 255 354)(200 355 256 386)(201 387 257 356)(202 357 258 388)(203 389 259 358)(204 359 260 390)(205 391 261 360)(206 361 262 392)(207 365 263 362)(208 363 264 366)(209 367 265 364)(210 337 266 368)(211 369 267 338)(212 339 268 370)(213 371 269 340)(214 341 270 372)(215 373 271 342)(216 343 272 374)(217 375 273 344)(218 345 274 376)(219 377 275 346)(220 347 276 378)(221 379 277 348)(222 349 278 380)(223 381 279 350)(224 351 280 382)

(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)

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

82 conjugacy classes

class 1 2A···2G4A···4F4G···4N7A7B7C14A···14U28A···28AJ
order12···24···44···477714···1428···28
size11···14···428···282222···24···4

82 irreducible representations

dim11111222222222244
type++++++-+-++-+-
imageC1C2C2C2C2D4Q8D4Q8D7C4○D4D14Dic14C7⋊D4C4○D28D4×D7Q8×D7
kernel(C2×C4)⋊Dic14C14.C42C2×Dic7⋊C4C14×C4⋊C4C22×Dic14C2×Dic7C2×Dic7C2×C28C2×C28C2×C4⋊C4C2×C14C22×C4C2×C4C2×C4C22C22C22
# reps13211442232912121266

Matrix representation of (C2×C4)⋊Dic14 ►in GL6(𝔽29)

2800000
0280000
001000
000100
0000280
0000028
,
100000
0280000
001000
000100
0000112
00002428
,
700000
0250000
0028100
0027100
0000315
00001326
,
010000
100000
007800
0012200
00002614
0000163

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,24,0,0,0,0,12,28],[7,0,0,0,0,0,0,25,0,0,0,0,0,0,28,27,0,0,0,0,1,1,0,0,0,0,0,0,3,13,0,0,0,0,15,26],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,1,0,0,0,0,8,22,0,0,0,0,0,0,26,16,0,0,0,0,14,3] >;
 

(C2×C4)⋊Dic14 in GAP, Magma, Sage, TeX 

(C_2\times C_4)\rtimes {\rm Dic}_{14}
 
% in TeX
 
G:=Group("(C2xC4):Dic14");
 
// GroupNames label
 
G:=SmallGroup(448,513);
 
// by ID
 
G=gap.SmallGroup(448,513);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,120,254,387,184,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^28=1,d^2=c^14,d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

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