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## G = C2×C52⋊3C4order 416 = 25·13

### Direct product of C2 and C52⋊3C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — C2×C52⋊3C4
 Chief series C1 — C13 — C26 — C2×C26 — C2×Dic13 — C22×Dic13 — C2×C52⋊3C4
 Lower central C13 — C26 — C2×C52⋊3C4
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×C523C4
G = < a,b,c | a2=b52=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 432 in 92 conjugacy classes, 65 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C22, C22 [×6], C2×C4 [×6], C2×C4 [×8], C23, C13, C4⋊C4 [×4], C22×C4, C22×C4 [×2], C26 [×3], C26 [×4], C2×C4⋊C4, Dic13 [×4], C52 [×4], C2×C26, C2×C26 [×6], C2×Dic13 [×4], C2×Dic13 [×4], C2×C52 [×6], C22×C26, C523C4 [×4], C22×Dic13 [×2], C22×C52, C2×C523C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D13, C2×C4⋊C4, Dic13 [×4], D26 [×3], Dic26 [×2], D52 [×2], C2×Dic13 [×6], C22×D13, C523C4 [×4], C2×Dic26, C2×D52, C22×Dic13, C2×C523C4

Smallest permutation representation of C2×C523C4
Regular action on 416 points
Generators in S416
(1 398)(2 399)(3 400)(4 401)(5 402)(6 403)(7 404)(8 405)(9 406)(10 407)(11 408)(12 409)(13 410)(14 411)(15 412)(16 413)(17 414)(18 415)(19 416)(20 365)(21 366)(22 367)(23 368)(24 369)(25 370)(26 371)(27 372)(28 373)(29 374)(30 375)(31 376)(32 377)(33 378)(34 379)(35 380)(36 381)(37 382)(38 383)(39 384)(40 385)(41 386)(42 387)(43 388)(44 389)(45 390)(46 391)(47 392)(48 393)(49 394)(50 395)(51 396)(52 397)(53 352)(54 353)(55 354)(56 355)(57 356)(58 357)(59 358)(60 359)(61 360)(62 361)(63 362)(64 363)(65 364)(66 313)(67 314)(68 315)(69 316)(70 317)(71 318)(72 319)(73 320)(74 321)(75 322)(76 323)(77 324)(78 325)(79 326)(80 327)(81 328)(82 329)(83 330)(84 331)(85 332)(86 333)(87 334)(88 335)(89 336)(90 337)(91 338)(92 339)(93 340)(94 341)(95 342)(96 343)(97 344)(98 345)(99 346)(100 347)(101 348)(102 349)(103 350)(104 351)(105 285)(106 286)(107 287)(108 288)(109 289)(110 290)(111 291)(112 292)(113 293)(114 294)(115 295)(116 296)(117 297)(118 298)(119 299)(120 300)(121 301)(122 302)(123 303)(124 304)(125 305)(126 306)(127 307)(128 308)(129 309)(130 310)(131 311)(132 312)(133 261)(134 262)(135 263)(136 264)(137 265)(138 266)(139 267)(140 268)(141 269)(142 270)(143 271)(144 272)(145 273)(146 274)(147 275)(148 276)(149 277)(150 278)(151 279)(152 280)(153 281)(154 282)(155 283)(156 284)(157 251)(158 252)(159 253)(160 254)(161 255)(162 256)(163 257)(164 258)(165 259)(166 260)(167 209)(168 210)(169 211)(170 212)(171 213)(172 214)(173 215)(174 216)(175 217)(176 218)(177 219)(178 220)(179 221)(180 222)(181 223)(182 224)(183 225)(184 226)(185 227)(186 228)(187 229)(188 230)(189 231)(190 232)(191 233)(192 234)(193 235)(194 236)(195 237)(196 238)(197 239)(198 240)(199 241)(200 242)(201 243)(202 244)(203 245)(204 246)(205 247)(206 248)(207 249)(208 250)
(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)
(1 259 150 330)(2 258 151 329)(3 257 152 328)(4 256 153 327)(5 255 154 326)(6 254 155 325)(7 253 156 324)(8 252 105 323)(9 251 106 322)(10 250 107 321)(11 249 108 320)(12 248 109 319)(13 247 110 318)(14 246 111 317)(15 245 112 316)(16 244 113 315)(17 243 114 314)(18 242 115 313)(19 241 116 364)(20 240 117 363)(21 239 118 362)(22 238 119 361)(23 237 120 360)(24 236 121 359)(25 235 122 358)(26 234 123 357)(27 233 124 356)(28 232 125 355)(29 231 126 354)(30 230 127 353)(31 229 128 352)(32 228 129 351)(33 227 130 350)(34 226 131 349)(35 225 132 348)(36 224 133 347)(37 223 134 346)(38 222 135 345)(39 221 136 344)(40 220 137 343)(41 219 138 342)(42 218 139 341)(43 217 140 340)(44 216 141 339)(45 215 142 338)(46 214 143 337)(47 213 144 336)(48 212 145 335)(49 211 146 334)(50 210 147 333)(51 209 148 332)(52 260 149 331)(53 376 187 308)(54 375 188 307)(55 374 189 306)(56 373 190 305)(57 372 191 304)(58 371 192 303)(59 370 193 302)(60 369 194 301)(61 368 195 300)(62 367 196 299)(63 366 197 298)(64 365 198 297)(65 416 199 296)(66 415 200 295)(67 414 201 294)(68 413 202 293)(69 412 203 292)(70 411 204 291)(71 410 205 290)(72 409 206 289)(73 408 207 288)(74 407 208 287)(75 406 157 286)(76 405 158 285)(77 404 159 284)(78 403 160 283)(79 402 161 282)(80 401 162 281)(81 400 163 280)(82 399 164 279)(83 398 165 278)(84 397 166 277)(85 396 167 276)(86 395 168 275)(87 394 169 274)(88 393 170 273)(89 392 171 272)(90 391 172 271)(91 390 173 270)(92 389 174 269)(93 388 175 268)(94 387 176 267)(95 386 177 266)(96 385 178 265)(97 384 179 264)(98 383 180 263)(99 382 181 262)(100 381 182 261)(101 380 183 312)(102 379 184 311)(103 378 185 310)(104 377 186 309)

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

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

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

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 1 2 2 2 2 2 2 2 2 type + + + + + - + - + + - + image C1 C2 C2 C2 C4 D4 Q8 D13 Dic13 D26 D26 Dic26 D52 kernel C2×C52⋊3C4 C52⋊3C4 C22×Dic13 C22×C52 C2×C52 C2×C26 C2×C26 C22×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 2 1 8 2 2 6 24 12 6 24 24

Matrix representation of C2×C523C4 in GL4(𝔽53) generated by

 52 0 0 0 0 1 0 0 0 0 52 0 0 0 0 52
,
 1 0 0 0 0 1 0 0 0 0 0 28 0 0 17 37
,
 1 0 0 0 0 23 0 0 0 0 9 49 0 0 20 44
G:=sub<GL(4,GF(53))| [52,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,0,17,0,0,28,37],[1,0,0,0,0,23,0,0,0,0,9,20,0,0,49,44] >;

C2×C523C4 in GAP, Magma, Sage, TeX

C_2\times C_{52}\rtimes_3C_4
% in TeX

G:=Group("C2xC52:3C4");
// GroupNames label

G:=SmallGroup(416,146);
// by ID

G=gap.SmallGroup(416,146);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,362,86,13829]);
// Polycyclic

G:=Group<a,b,c|a^2=b^52=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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