C2^2xDic28 - GroupNames

G = C₂²×Dic₂₈ order 448 = 2⁶·7

Direct product of C₂² and Dic₂₈

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂²×Dic₂₈, C₅₆.₆₁C₂³, C₂₈.₅₇C₂⁴, C₂³.₆₂D₂₈, Dic₁₄.₂₁C₂³, (C₂×C₁₄)⋊₆Q₁₆, C₁₄⋊₁(C₂×Q₁₆), C₇⋊₁(C₂²×Q₁₆), C₄.₄₇(C₂×D₂₈), C₂₈.₂₉₂(C₂×D₄), (C₂×C₄).₁₀₂D₂₈, (C₂×C₈).₃₁₀D₁₄, (C₂×C₂₈).₃₉₂D₄, C₈.₅₂(C₂²×D₇), C₄.₅₄(C₂³×D₇), (C₂²×C₈).₁₀D₇, (C₂²×C₅₆).₁₆C₂, C₂².₇₂(C₂×D₂₈), C₂.₂₆(C₂²×D₂₈), C₁₄.₂₄(C₂²×D₄), (C₂×C₅₆).₃₈₂C₂², (C₂×C₂₈).₇₈₈C₂³, (C₂²×C₄).₄₄₅D₁₄, (C₂²×C₁₄).₁₄₇D₄, (C₂²×Dic₁₄).₉C₂, (C₂²×C₂₈).₅₂₇C₂², (C₂×Dic₁₄).₂₅₈C₂², (C₂×C₁₄).₁₈₀(C₂×D₄), (C₂×C₄).₇₃₈(C₂²×D₇), SmallGroup(448,1195)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — C₂²×Dic₂₈

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — Dic₁₄ — C₂×Dic₁₄ — C₂²×Dic₁₄ — C₂²×Dic₂₈

Lower central

C₇ — C₁₄ — C₂₈ — C₂²×Dic₂₈

Upper central

C₁ — C₂³ — C₂²×C₄ — C₂²×C₈

Generators and relations for C₂²×Dic₂₈
G = < a,b,c,d | a²=b²=c⁵⁶=1, d²=c²⁸, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 1124 in 258 conjugacy classes, 127 normal (13 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₇, C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₄, C₁₄, C₂×C₈, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×Q₈, Dic₇, C₂₈, C₂₈, C₂×C₁₄, C₂²×C₈, C₂×Q₁₆, C₂²×Q₈, C₅₆, Dic₁₄, Dic₁₄, C₂×Dic₇, C₂×C₂₈, C₂²×C₁₄, C₂²×Q₁₆, Dic₂₈, C₂×C₅₆, C₂×Dic₁₄, C₂×Dic₁₄, C₂²×Dic₇, C₂²×C₂₈, C₂×Dic₂₈, C₂²×C₅₆, C₂²×Dic₁₄, C₂²×Dic₂₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, Q₁₆, C₂×D₄, C₂⁴, D₁₄, C₂×Q₁₆, C₂²×D₄, D₂₈, C₂²×D₇, C₂²×Q₁₆, Dic₂₈, C₂×D₂₈, C₂³×D₇, C₂×Dic₂₈, C₂²×D₂₈, C₂²×Dic₂₈

Smallest permutation representation of C₂²×Dic₂₈
►Regular action on 448 points

Generators in S₄₄₈

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

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

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

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

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

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

124 conjugacy classes

class	1	2A	···	2G	4A	4B	4C	4D	4E	···	4L	7A	7B	7C	8A	···	8H	14A	···	14U	28A	···	28X	56A	···	56AV
order	1	2	···	2	4	4	4	4	4	···	4	7	7	7	8	···	8	14	···	14	28	···	28	56	···	56
size	1	1	···	1	2	2	2	2	28	···	28	2	2	2	2	···	2	2	···	2	2	···	2	2	···	2

124 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	2	2
type	+	+	+	+	+	+	+	-	+	+	+	+	-
image	C₁	C₂	C₂	C₂	D₄	D₄	D₇	Q₁₆	D₁₄	D₁₄	D₂₈	D₂₈	Dic₂₈
kernel	C₂²×Dic₂₈	C₂×Dic₂₈	C₂²×C₅₆	C₂²×Dic₁₄	C₂×C₂₈	C₂²×C₁₄	C₂²×C₈	C₂×C₁₄	C₂×C₈	C₂²×C₄	C₂×C₄	C₂³	C₂²
# reps	1	12	1	2	3	1	3	8	18	3	18	6	48

Matrix representation of C₂²×Dic₂₈ ►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	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	46	104	0	0
0	9	13	0	0
0	0	0	0	62
0	0	0	82	62

112	0	0	0	0
0	9	67	0	0
0	100	104	0	0
0	0	0	60	3
0	0	0	5	53

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],[112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,112],[112,0,0,0,0,0,46,9,0,0,0,104,13,0,0,0,0,0,0,82,0,0,0,62,62],[112,0,0,0,0,0,9,100,0,0,0,67,104,0,0,0,0,0,60,5,0,0,0,3,53] >;

C₂²×Dic₂₈ in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{28}

% in TeX

G:=Group("C2^2xDic28");

// GroupNames label

G:=SmallGroup(448,1195);

// by ID

G=gap.SmallGroup(448,1195);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,675,192,1684,102,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^56=1,d^2=c^28,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;

// generators/relations

G = C22×Dic28 order 448 = 26·7

Direct product of C22 and Dic28

G = C₂²×Dic₂₈ order 448 = 2⁶·7

Direct product of C₂² and Dic₂₈