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## G = C2×C4×Dic13order 416 = 25·13

### Direct product of C2×C4 and Dic13

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C4×Dic13
G = < a,b,c,d | a2=b4=c26=1, d2=c13, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 432 in 108 conjugacy classes, 81 normal (11 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×8], C22, C22 [×6], C2×C4 [×6], C2×C4 [×12], C23, C13, C42 [×4], C22×C4, C22×C4 [×2], C26, C26 [×6], C2×C42, Dic13 [×8], C52 [×4], C2×C26, C2×C26 [×6], C2×Dic13 [×12], C2×C52 [×6], C22×C26, C4×Dic13 [×4], C22×Dic13 [×2], C22×C52, C2×C4×Dic13
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], D13, C2×C42, Dic13 [×4], D26 [×3], C4×D13 [×4], C2×Dic13 [×6], C22×D13, C4×Dic13 [×4], C2×C4×D13 [×2], C22×Dic13, C2×C4×Dic13

Smallest permutation representation of C2×C4×Dic13
Regular action on 416 points
Generators in S416
(1 356)(2 357)(3 358)(4 359)(5 360)(6 361)(7 362)(8 363)(9 364)(10 339)(11 340)(12 341)(13 342)(14 343)(15 344)(16 345)(17 346)(18 347)(19 348)(20 349)(21 350)(22 351)(23 352)(24 353)(25 354)(26 355)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 65)(40 66)(41 67)(42 68)(43 69)(44 70)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 77)(52 78)(79 149)(80 150)(81 151)(82 152)(83 153)(84 154)(85 155)(86 156)(87 131)(88 132)(89 133)(90 134)(91 135)(92 136)(93 137)(94 138)(95 139)(96 140)(97 141)(98 142)(99 143)(100 144)(101 145)(102 146)(103 147)(104 148)(105 252)(106 253)(107 254)(108 255)(109 256)(110 257)(111 258)(112 259)(113 260)(114 235)(115 236)(116 237)(117 238)(118 239)(119 240)(120 241)(121 242)(122 243)(123 244)(124 245)(125 246)(126 247)(127 248)(128 249)(129 250)(130 251)(157 329)(158 330)(159 331)(160 332)(161 333)(162 334)(163 335)(164 336)(165 337)(166 338)(167 313)(168 314)(169 315)(170 316)(171 317)(172 318)(173 319)(174 320)(175 321)(176 322)(177 323)(178 324)(179 325)(180 326)(181 327)(182 328)(183 308)(184 309)(185 310)(186 311)(187 312)(188 287)(189 288)(190 289)(191 290)(192 291)(193 292)(194 293)(195 294)(196 295)(197 296)(198 297)(199 298)(200 299)(201 300)(202 301)(203 302)(204 303)(205 304)(206 305)(207 306)(208 307)(209 407)(210 408)(211 409)(212 410)(213 411)(214 412)(215 413)(216 414)(217 415)(218 416)(219 391)(220 392)(221 393)(222 394)(223 395)(224 396)(225 397)(226 398)(227 399)(228 400)(229 401)(230 402)(231 403)(232 404)(233 405)(234 406)(261 386)(262 387)(263 388)(264 389)(265 390)(266 365)(267 366)(268 367)(269 368)(270 369)(271 370)(272 371)(273 372)(274 373)(275 374)(276 375)(277 376)(278 377)(279 378)(280 379)(281 380)(282 381)(283 382)(284 383)(285 384)(286 385)
(1 232 52 327)(2 233 27 328)(3 234 28 329)(4 209 29 330)(5 210 30 331)(6 211 31 332)(7 212 32 333)(8 213 33 334)(9 214 34 335)(10 215 35 336)(11 216 36 337)(12 217 37 338)(13 218 38 313)(14 219 39 314)(15 220 40 315)(16 221 41 316)(17 222 42 317)(18 223 43 318)(19 224 44 319)(20 225 45 320)(21 226 46 321)(22 227 47 322)(23 228 48 323)(24 229 49 324)(25 230 50 325)(26 231 51 326)(53 182 357 405)(54 157 358 406)(55 158 359 407)(56 159 360 408)(57 160 361 409)(58 161 362 410)(59 162 363 411)(60 163 364 412)(61 164 339 413)(62 165 340 414)(63 166 341 415)(64 167 342 416)(65 168 343 391)(66 169 344 392)(67 170 345 393)(68 171 346 394)(69 172 347 395)(70 173 348 396)(71 174 349 397)(72 175 350 398)(73 176 351 399)(74 177 352 400)(75 178 353 401)(76 179 354 402)(77 180 355 403)(78 181 356 404)(79 118 367 189)(80 119 368 190)(81 120 369 191)(82 121 370 192)(83 122 371 193)(84 123 372 194)(85 124 373 195)(86 125 374 196)(87 126 375 197)(88 127 376 198)(89 128 377 199)(90 129 378 200)(91 130 379 201)(92 105 380 202)(93 106 381 203)(94 107 382 204)(95 108 383 205)(96 109 384 206)(97 110 385 207)(98 111 386 208)(99 112 387 183)(100 113 388 184)(101 114 389 185)(102 115 390 186)(103 116 365 187)(104 117 366 188)(131 247 276 296)(132 248 277 297)(133 249 278 298)(134 250 279 299)(135 251 280 300)(136 252 281 301)(137 253 282 302)(138 254 283 303)(139 255 284 304)(140 256 285 305)(141 257 286 306)(142 258 261 307)(143 259 262 308)(144 260 263 309)(145 235 264 310)(146 236 265 311)(147 237 266 312)(148 238 267 287)(149 239 268 288)(150 240 269 289)(151 241 270 290)(152 242 271 291)(153 243 272 292)(154 244 273 293)(155 245 274 294)(156 246 275 295)
(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)
(1 144 14 131)(2 143 15 156)(3 142 16 155)(4 141 17 154)(5 140 18 153)(6 139 19 152)(7 138 20 151)(8 137 21 150)(9 136 22 149)(10 135 23 148)(11 134 24 147)(12 133 25 146)(13 132 26 145)(27 262 40 275)(28 261 41 274)(29 286 42 273)(30 285 43 272)(31 284 44 271)(32 283 45 270)(33 282 46 269)(34 281 47 268)(35 280 48 267)(36 279 49 266)(37 278 50 265)(38 277 51 264)(39 276 52 263)(53 387 66 374)(54 386 67 373)(55 385 68 372)(56 384 69 371)(57 383 70 370)(58 382 71 369)(59 381 72 368)(60 380 73 367)(61 379 74 366)(62 378 75 365)(63 377 76 390)(64 376 77 389)(65 375 78 388)(79 364 92 351)(80 363 93 350)(81 362 94 349)(82 361 95 348)(83 360 96 347)(84 359 97 346)(85 358 98 345)(86 357 99 344)(87 356 100 343)(88 355 101 342)(89 354 102 341)(90 353 103 340)(91 352 104 339)(105 399 118 412)(106 398 119 411)(107 397 120 410)(108 396 121 409)(109 395 122 408)(110 394 123 407)(111 393 124 406)(112 392 125 405)(113 391 126 404)(114 416 127 403)(115 415 128 402)(116 414 129 401)(117 413 130 400)(157 208 170 195)(158 207 171 194)(159 206 172 193)(160 205 173 192)(161 204 174 191)(162 203 175 190)(163 202 176 189)(164 201 177 188)(165 200 178 187)(166 199 179 186)(167 198 180 185)(168 197 181 184)(169 196 182 183)(209 257 222 244)(210 256 223 243)(211 255 224 242)(212 254 225 241)(213 253 226 240)(214 252 227 239)(215 251 228 238)(216 250 229 237)(217 249 230 236)(218 248 231 235)(219 247 232 260)(220 246 233 259)(221 245 234 258)(287 336 300 323)(288 335 301 322)(289 334 302 321)(290 333 303 320)(291 332 304 319)(292 331 305 318)(293 330 306 317)(294 329 307 316)(295 328 308 315)(296 327 309 314)(297 326 310 313)(298 325 311 338)(299 324 312 337)

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

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

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

128 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4X 13A ··· 13F 26A ··· 26AP 52A ··· 52AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 13 ··· 13 26 ··· 26 52 ··· 52 size 1 1 ··· 1 1 ··· 1 13 ··· 13 2 ··· 2 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + - + + image C1 C2 C2 C2 C4 C4 D13 Dic13 D26 D26 C4×D13 kernel C2×C4×Dic13 C4×Dic13 C22×Dic13 C22×C52 C2×Dic13 C2×C52 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 16 8 6 24 12 6 48

Matrix representation of C2×C4×Dic13 in GL4(𝔽53) generated by

 1 0 0 0 0 52 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 52 0 0 0 0 23 0 0 0 0 23
,
 52 0 0 0 0 1 0 0 0 0 0 1 0 0 52 11
,
 30 0 0 0 0 52 0 0 0 0 16 14 0 0 31 37
G:=sub<GL(4,GF(53))| [1,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,52,0,0,0,0,23,0,0,0,0,23],[52,0,0,0,0,1,0,0,0,0,0,52,0,0,1,11],[30,0,0,0,0,52,0,0,0,0,16,31,0,0,14,37] >;

C2×C4×Dic13 in GAP, Magma, Sage, TeX

C_2\times C_4\times {\rm Dic}_{13}
% in TeX

G:=Group("C2xC4xDic13");
// GroupNames label

G:=SmallGroup(416,143);
// by ID

G=gap.SmallGroup(416,143);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,86,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^26=1,d^2=c^13,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