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G = C22×Dic28  order 448 = 26·7

Direct product of C22 and Dic28

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

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

Subgroups: 1124 in 258 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C4 [×8], C22 [×7], C7, C8 [×4], C2×C4 [×6], C2×C4 [×12], Q8 [×20], C23, C14, C14 [×6], C2×C8 [×6], Q16 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×18], Dic7 [×8], C28, C28 [×3], C2×C14 [×7], C22×C8, C2×Q16 [×12], C22×Q8 [×2], C56 [×4], Dic14 [×8], Dic14 [×12], C2×Dic7 [×12], C2×C28 [×6], C22×C14, C22×Q16, Dic28 [×16], C2×C56 [×6], C2×Dic14 [×12], C2×Dic14 [×6], C22×Dic7 [×2], C22×C28, C2×Dic28 [×12], C22×C56, C22×Dic14 [×2], C22×Dic28

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, Q16 [×4], C2×D4 [×6], C24, D14 [×7], C2×Q16 [×6], C22×D4, D28 [×4], C22×D7 [×7], C22×Q16, Dic28 [×4], C2×D28 [×6], C23×D7, C2×Dic28 [×6], C22×D28, C22×Dic28

Generators and relations
 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 >

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

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

(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)

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

Matrix representation ►G ⊆ GL5(𝔽113)

10000
0112000
0011200
0001120
0000112
,
1120000
0112000
0011200
0001120
0000112
,
1120000
04610400
091300
000062
0008262
,
1120000
096700
010010400
000603
000553

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] >;
 

124 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4L7A7B7C8A···8H14A···14U28A···28X56A···56AV
order12···244444···47778···814···1428···2856···56
size11···1222228···282222···22···22···22···2

124 irreducible representations

dim1111222222222
type+++++++-++++-
imageC1C2C2C2D4D4D7Q16D14D14D28D28Dic28
kernelC22×Dic28C2×Dic28C22×C56C22×Dic14C2×C28C22×C14C22×C8C2×C14C2×C8C22×C4C2×C4C23C22
# reps11212313818318648

In GAP, Magma, Sage, TeX 

C_2^2\times 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
 

׿
×
⋊
ℤ
𝔽
○
≀