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## G = C2×Dic52order 416 = 25·13

### Direct product of C2 and Dic52

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×Dic52
 Chief series C1 — C13 — C26 — C52 — Dic26 — C2×Dic26 — C2×Dic52
 Lower central C13 — C26 — C52 — C2×Dic52
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×Dic52
G = < a,b,c | a2=b104=1, c2=b52, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 400 in 60 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C8, C2×C4, C2×C4, Q8, C13, C2×C8, Q16, C2×Q8, C26, C26, C2×Q16, Dic13, C52, C2×C26, C104, Dic26, Dic26, C2×Dic13, C2×C52, Dic52, C2×C104, C2×Dic26, C2×Dic52
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, D13, C2×Q16, D26, D52, C22×D13, Dic52, C2×D52, C2×Dic52

Smallest permutation representation of C2×Dic52
Regular action on 416 points
Generators in S416
(1 160)(2 161)(3 162)(4 163)(5 164)(6 165)(7 166)(8 167)(9 168)(10 169)(11 170)(12 171)(13 172)(14 173)(15 174)(16 175)(17 176)(18 177)(19 178)(20 179)(21 180)(22 181)(23 182)(24 183)(25 184)(26 185)(27 186)(28 187)(29 188)(30 189)(31 190)(32 191)(33 192)(34 193)(35 194)(36 195)(37 196)(38 197)(39 198)(40 199)(41 200)(42 201)(43 202)(44 203)(45 204)(46 205)(47 206)(48 207)(49 208)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 120)(66 121)(67 122)(68 123)(69 124)(70 125)(71 126)(72 127)(73 128)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 155)(101 156)(102 157)(103 158)(104 159)(209 324)(210 325)(211 326)(212 327)(213 328)(214 329)(215 330)(216 331)(217 332)(218 333)(219 334)(220 335)(221 336)(222 337)(223 338)(224 339)(225 340)(226 341)(227 342)(228 343)(229 344)(230 345)(231 346)(232 347)(233 348)(234 349)(235 350)(236 351)(237 352)(238 353)(239 354)(240 355)(241 356)(242 357)(243 358)(244 359)(245 360)(246 361)(247 362)(248 363)(249 364)(250 365)(251 366)(252 367)(253 368)(254 369)(255 370)(256 371)(257 372)(258 373)(259 374)(260 375)(261 376)(262 377)(263 378)(264 379)(265 380)(266 381)(267 382)(268 383)(269 384)(270 385)(271 386)(272 387)(273 388)(274 389)(275 390)(276 391)(277 392)(278 393)(279 394)(280 395)(281 396)(282 397)(283 398)(284 399)(285 400)(286 401)(287 402)(288 403)(289 404)(290 405)(291 406)(292 407)(293 408)(294 409)(295 410)(296 411)(297 412)(298 413)(299 414)(300 415)(301 416)(302 313)(303 314)(304 315)(305 316)(306 317)(307 318)(308 319)(309 320)(310 321)(311 322)(312 323)
(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)
(1 378 53 326)(2 377 54 325)(3 376 55 324)(4 375 56 323)(5 374 57 322)(6 373 58 321)(7 372 59 320)(8 371 60 319)(9 370 61 318)(10 369 62 317)(11 368 63 316)(12 367 64 315)(13 366 65 314)(14 365 66 313)(15 364 67 416)(16 363 68 415)(17 362 69 414)(18 361 70 413)(19 360 71 412)(20 359 72 411)(21 358 73 410)(22 357 74 409)(23 356 75 408)(24 355 76 407)(25 354 77 406)(26 353 78 405)(27 352 79 404)(28 351 80 403)(29 350 81 402)(30 349 82 401)(31 348 83 400)(32 347 84 399)(33 346 85 398)(34 345 86 397)(35 344 87 396)(36 343 88 395)(37 342 89 394)(38 341 90 393)(39 340 91 392)(40 339 92 391)(41 338 93 390)(42 337 94 389)(43 336 95 388)(44 335 96 387)(45 334 97 386)(46 333 98 385)(47 332 99 384)(48 331 100 383)(49 330 101 382)(50 329 102 381)(51 328 103 380)(52 327 104 379)(105 214 157 266)(106 213 158 265)(107 212 159 264)(108 211 160 263)(109 210 161 262)(110 209 162 261)(111 312 163 260)(112 311 164 259)(113 310 165 258)(114 309 166 257)(115 308 167 256)(116 307 168 255)(117 306 169 254)(118 305 170 253)(119 304 171 252)(120 303 172 251)(121 302 173 250)(122 301 174 249)(123 300 175 248)(124 299 176 247)(125 298 177 246)(126 297 178 245)(127 296 179 244)(128 295 180 243)(129 294 181 242)(130 293 182 241)(131 292 183 240)(132 291 184 239)(133 290 185 238)(134 289 186 237)(135 288 187 236)(136 287 188 235)(137 286 189 234)(138 285 190 233)(139 284 191 232)(140 283 192 231)(141 282 193 230)(142 281 194 229)(143 280 195 228)(144 279 196 227)(145 278 197 226)(146 277 198 225)(147 276 199 224)(148 275 200 223)(149 274 201 222)(150 273 202 221)(151 272 203 220)(152 271 204 219)(153 270 205 218)(154 269 206 217)(155 268 207 216)(156 267 208 215)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 13A ··· 13F 26A ··· 26R 52A ··· 52X 104A ··· 104AV order 1 2 2 2 4 4 4 4 4 4 8 8 8 8 13 ··· 13 26 ··· 26 52 ··· 52 104 ··· 104 size 1 1 1 1 2 2 52 52 52 52 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

110 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + - + + + + + - image C1 C2 C2 C2 D4 D4 Q16 D13 D26 D26 D52 D52 Dic52 kernel C2×Dic52 Dic52 C2×C104 C2×Dic26 C52 C2×C26 C26 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 2 1 1 4 6 12 6 12 12 48

Matrix representation of C2×Dic52 in GL4(𝔽313) generated by

 312 0 0 0 0 312 0 0 0 0 312 0 0 0 0 312
,
 0 1 0 0 312 198 0 0 0 0 0 134 0 0 7 193
,
 240 259 0 0 203 73 0 0 0 0 35 149 0 0 80 278
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,312,0,0,0,0,312],[0,312,0,0,1,198,0,0,0,0,0,7,0,0,134,193],[240,203,0,0,259,73,0,0,0,0,35,80,0,0,149,278] >;

C2×Dic52 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{52}
% in TeX

G:=Group("C2xDic52");
// GroupNames label

G:=SmallGroup(416,126);
// by ID

G=gap.SmallGroup(416,126);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,218,122,579,69,13829]);
// Polycyclic

G:=Group<a,b,c|a^2=b^104=1,c^2=b^52,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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