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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 [×6], C4 [×4], C4 [×8], C22 [×7], C2×C4 [×6], C2×C4 [×12], Q8 [×16], C23, C13, C22×C4, C22×C4 [×2], C2×Q8 [×12], C26, C26 [×6], C22×Q8, Dic13 [×8], C52 [×4], C2×C26 [×7], Dic26 [×16], C2×Dic13 [×12], C2×C52 [×6], C22×C26, C2×Dic26 [×12], C22×Dic13 [×2], C22×C52, C22×Dic26
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C24, D13, C22×Q8, D26 [×7], Dic26 [×4], C22×D13 [×7], C2×Dic26 [×6], C23×D13, C22×Dic26

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

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

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

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

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