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## G = C22×Dic26order 416 = 25·13

### Direct product of C22 and Dic26

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C22×Dic26
 Chief series C1 — C13 — C26 — Dic13 — C2×Dic13 — C22×Dic13 — C22×Dic26
 Lower central C13 — C26 — C22×Dic26
 Upper central C1 — C23 — C22×C4

Generators and relations for C22×Dic26
G = < a,b,c,d | a2=b2=c52=1, d2=c26, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 768 in 156 conjugacy classes, 105 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C23, C13, C22×C4, C22×C4, C2×Q8, C26, C26, C22×Q8, Dic13, C52, C2×C26, Dic26, C2×Dic13, C2×C52, C22×C26, C2×Dic26, C22×Dic13, C22×C52, C22×Dic26
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, D13, C22×Q8, D26, Dic26, C22×D13, C2×Dic26, C23×D13, C22×Dic26

Smallest permutation representation of C22×Dic26
Regular action on 416 points
Generators in S416
(1 124)(2 125)(3 126)(4 127)(5 128)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 140)(18 141)(19 142)(20 143)(21 144)(22 145)(23 146)(24 147)(25 148)(26 149)(27 150)(28 151)(29 152)(30 153)(31 154)(32 155)(33 156)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 121)(51 122)(52 123)(53 293)(54 294)(55 295)(56 296)(57 297)(58 298)(59 299)(60 300)(61 301)(62 302)(63 303)(64 304)(65 305)(66 306)(67 307)(68 308)(69 309)(70 310)(71 311)(72 312)(73 261)(74 262)(75 263)(76 264)(77 265)(78 266)(79 267)(80 268)(81 269)(82 270)(83 271)(84 272)(85 273)(86 274)(87 275)(88 276)(89 277)(90 278)(91 279)(92 280)(93 281)(94 282)(95 283)(96 284)(97 285)(98 286)(99 287)(100 288)(101 289)(102 290)(103 291)(104 292)(157 232)(158 233)(159 234)(160 235)(161 236)(162 237)(163 238)(164 239)(165 240)(166 241)(167 242)(168 243)(169 244)(170 245)(171 246)(172 247)(173 248)(174 249)(175 250)(176 251)(177 252)(178 253)(179 254)(180 255)(181 256)(182 257)(183 258)(184 259)(185 260)(186 209)(187 210)(188 211)(189 212)(190 213)(191 214)(192 215)(193 216)(194 217)(195 218)(196 219)(197 220)(198 221)(199 222)(200 223)(201 224)(202 225)(203 226)(204 227)(205 228)(206 229)(207 230)(208 231)(313 368)(314 369)(315 370)(316 371)(317 372)(318 373)(319 374)(320 375)(321 376)(322 377)(323 378)(324 379)(325 380)(326 381)(327 382)(328 383)(329 384)(330 385)(331 386)(332 387)(333 388)(334 389)(335 390)(336 391)(337 392)(338 393)(339 394)(340 395)(341 396)(342 397)(343 398)(344 399)(345 400)(346 401)(347 402)(348 403)(349 404)(350 405)(351 406)(352 407)(353 408)(354 409)(355 410)(356 411)(357 412)(358 413)(359 414)(360 415)(361 416)(362 365)(363 366)(364 367)
(1 416)(2 365)(3 366)(4 367)(5 368)(6 369)(7 370)(8 371)(9 372)(10 373)(11 374)(12 375)(13 376)(14 377)(15 378)(16 379)(17 380)(18 381)(19 382)(20 383)(21 384)(22 385)(23 386)(24 387)(25 388)(26 389)(27 390)(28 391)(29 392)(30 393)(31 394)(32 395)(33 396)(34 397)(35 398)(36 399)(37 400)(38 401)(39 402)(40 403)(41 404)(42 405)(43 406)(44 407)(45 408)(46 409)(47 410)(48 411)(49 412)(50 413)(51 414)(52 415)(53 185)(54 186)(55 187)(56 188)(57 189)(58 190)(59 191)(60 192)(61 193)(62 194)(63 195)(64 196)(65 197)(66 198)(67 199)(68 200)(69 201)(70 202)(71 203)(72 204)(73 205)(74 206)(75 207)(76 208)(77 157)(78 158)(79 159)(80 160)(81 161)(82 162)(83 163)(84 164)(85 165)(86 166)(87 167)(88 168)(89 169)(90 170)(91 171)(92 172)(93 173)(94 174)(95 175)(96 176)(97 177)(98 178)(99 179)(100 180)(101 181)(102 182)(103 183)(104 184)(105 342)(106 343)(107 344)(108 345)(109 346)(110 347)(111 348)(112 349)(113 350)(114 351)(115 352)(116 353)(117 354)(118 355)(119 356)(120 357)(121 358)(122 359)(123 360)(124 361)(125 362)(126 363)(127 364)(128 313)(129 314)(130 315)(131 316)(132 317)(133 318)(134 319)(135 320)(136 321)(137 322)(138 323)(139 324)(140 325)(141 326)(142 327)(143 328)(144 329)(145 330)(146 331)(147 332)(148 333)(149 334)(150 335)(151 336)(152 337)(153 338)(154 339)(155 340)(156 341)(209 294)(210 295)(211 296)(212 297)(213 298)(214 299)(215 300)(216 301)(217 302)(218 303)(219 304)(220 305)(221 306)(222 307)(223 308)(224 309)(225 310)(226 311)(227 312)(228 261)(229 262)(230 263)(231 264)(232 265)(233 266)(234 267)(235 268)(236 269)(237 270)(238 271)(239 272)(240 273)(241 274)(242 275)(243 276)(244 277)(245 278)(246 279)(247 280)(248 281)(249 282)(250 283)(251 284)(252 285)(253 286)(254 287)(255 288)(256 289)(257 290)(258 291)(259 292)(260 293)
(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)
(1 60 27 86)(2 59 28 85)(3 58 29 84)(4 57 30 83)(5 56 31 82)(6 55 32 81)(7 54 33 80)(8 53 34 79)(9 104 35 78)(10 103 36 77)(11 102 37 76)(12 101 38 75)(13 100 39 74)(14 99 40 73)(15 98 41 72)(16 97 42 71)(17 96 43 70)(18 95 44 69)(19 94 45 68)(20 93 46 67)(21 92 47 66)(22 91 48 65)(23 90 49 64)(24 89 50 63)(25 88 51 62)(26 87 52 61)(105 267 131 293)(106 266 132 292)(107 265 133 291)(108 264 134 290)(109 263 135 289)(110 262 136 288)(111 261 137 287)(112 312 138 286)(113 311 139 285)(114 310 140 284)(115 309 141 283)(116 308 142 282)(117 307 143 281)(118 306 144 280)(119 305 145 279)(120 304 146 278)(121 303 147 277)(122 302 148 276)(123 301 149 275)(124 300 150 274)(125 299 151 273)(126 298 152 272)(127 297 153 271)(128 296 154 270)(129 295 155 269)(130 294 156 268)(157 373 183 399)(158 372 184 398)(159 371 185 397)(160 370 186 396)(161 369 187 395)(162 368 188 394)(163 367 189 393)(164 366 190 392)(165 365 191 391)(166 416 192 390)(167 415 193 389)(168 414 194 388)(169 413 195 387)(170 412 196 386)(171 411 197 385)(172 410 198 384)(173 409 199 383)(174 408 200 382)(175 407 201 381)(176 406 202 380)(177 405 203 379)(178 404 204 378)(179 403 205 377)(180 402 206 376)(181 401 207 375)(182 400 208 374)(209 341 235 315)(210 340 236 314)(211 339 237 313)(212 338 238 364)(213 337 239 363)(214 336 240 362)(215 335 241 361)(216 334 242 360)(217 333 243 359)(218 332 244 358)(219 331 245 357)(220 330 246 356)(221 329 247 355)(222 328 248 354)(223 327 249 353)(224 326 250 352)(225 325 251 351)(226 324 252 350)(227 323 253 349)(228 322 254 348)(229 321 255 347)(230 320 256 346)(231 319 257 345)(232 318 258 344)(233 317 259 343)(234 316 260 342)

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

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

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

116 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 13A ··· 13F 26A ··· 26AP 52A ··· 52AV order 1 2 ··· 2 4 4 4 4 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 ··· 1 2 2 2 2 26 ··· 26 2 ··· 2 2 ··· 2 2 ··· 2

116 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + - + + + - image C1 C2 C2 C2 Q8 D13 D26 D26 Dic26 kernel C22×Dic26 C2×Dic26 C22×Dic13 C22×C52 C2×C26 C22×C4 C2×C4 C23 C22 # reps 1 12 2 1 4 6 36 6 48

Matrix representation of C22×Dic26 in GL4(𝔽53) generated by

 52 0 0 0 0 52 0 0 0 0 52 0 0 0 0 52
,
 1 0 0 0 0 52 0 0 0 0 52 0 0 0 0 52
,
 52 0 0 0 0 1 0 0 0 0 49 35 0 0 21 28
,
 1 0 0 0 0 1 0 0 0 0 31 35 0 0 24 22
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[1,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[52,0,0,0,0,1,0,0,0,0,49,21,0,0,35,28],[1,0,0,0,0,1,0,0,0,0,31,24,0,0,35,22] >;

C22×Dic26 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{26}
% in TeX

G:=Group("C2^2xDic26");
// GroupNames label

G:=SmallGroup(416,212);
// by ID

G=gap.SmallGroup(416,212);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,579,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^52=1,d^2=c^26,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

׿
×
𝔽