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G = C2×Dic52order 416 = 25·13

Direct product of C2 and Dic52

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

Aliases: C2×Dic52, C261Q16, C4.8D52, C8.16D26, C52.31D4, C52.31C23, C22.14D52, C104.18C22, Dic26.7C22, C131(C2×Q16), (C2×C8).4D13, (C2×C104).6C2, C26.12(C2×D4), C2.14(C2×D52), (C2×C4).82D26, (C2×C26).19D4, (C2×C52).90C22, (C2×Dic26).5C2, C4.29(C22×D13), SmallGroup(416,126)

Series: Derived Chief Lower central Upper central

C1C52 — C2×Dic52
C1C13C26C52Dic26C2×Dic26 — C2×Dic52
C13C26C52 — C2×Dic52
C1C22C2×C4C2×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 [×2], C4 [×2], C4 [×4], C22, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×6], C13, C2×C8, Q16 [×4], C2×Q8 [×2], C26, C26 [×2], C2×Q16, Dic13 [×4], C52 [×2], C2×C26, C104 [×2], Dic26 [×4], Dic26 [×2], C2×Dic13 [×2], C2×C52, Dic52 [×4], C2×C104, C2×Dic26 [×2], C2×Dic52
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, Q16 [×2], C2×D4, D13, C2×Q16, D26 [×3], D52 [×2], C22×D13, Dic52 [×2], C2×D52, C2×Dic52

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

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

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

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

110 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F8A8B8C8D13A···13F26A···26R52A···52X104A···104AV
order1222444444888813···1326···2652···52104···104
size1111225252525222222···22···22···22···2

110 irreducible representations

dim1111222222222
type++++++-+++++-
imageC1C2C2C2D4D4Q16D13D26D26D52D52Dic52
kernelC2×Dic52Dic52C2×C104C2×Dic26C52C2×C26C26C2×C8C8C2×C4C4C22C2
# reps14121146126121248

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

312000
031200
003120
000312
,
0100
31219800
000134
007193
,
24025900
2037300
0035149
0080278
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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