metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C26.10C42, C22.11D52, C23.27D26, C22.3Dic26, (C2×C52)⋊8C4, (C2×C26).4Q8, (C2×C4)⋊2Dic13, (C2×C26).32D4, C26.16(C4⋊C4), (C2×Dic13)⋊3C4, (C22×C52).1C2, C2.5(C4×Dic13), C2.2(C52⋊3C4), (C22×C4).2D13, C22.12(C4×D13), C2.2(D26⋊C4), C26.22(C22⋊C4), C13⋊2(C2.C42), C2.2(C26.D4), C2.2(C23.D13), C22.16(C13⋊D4), (C22×C26).31C22, (C22×Dic13).1C2, C22.10(C2×Dic13), (C2×C26).33(C2×C4), SmallGroup(416,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C26.10C42
G = < a,b,c | a26=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=a13b >
Subgroups: 376 in 76 conjugacy classes, 45 normal (19 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C13, C22×C4, C22×C4, C26, C26, C2.C42, Dic13, C52, C2×C26, C2×C26, C2×Dic13, C2×Dic13, C2×C52, C2×C52, C22×C26, C22×Dic13, C22×C52, C26.10C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, D13, C2.C42, Dic13, D26, Dic26, C4×D13, D52, C2×Dic13, C13⋊D4, C4×Dic13, C26.D4, C52⋊3C4, D26⋊C4, C23.D13, C26.10C42
►Regular action on 416 points
Generators in S416
(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)
(1 282 59 230)(2 281 60 229)(3 280 61 228)(4 279 62 227)(5 278 63 226)(6 277 64 225)(7 276 65 224)(8 275 66 223)(9 274 67 222)(10 273 68 221)(11 272 69 220)(12 271 70 219)(13 270 71 218)(14 269 72 217)(15 268 73 216)(16 267 74 215)(17 266 75 214)(18 265 76 213)(19 264 77 212)(20 263 78 211)(21 262 53 210)(22 261 54 209)(23 286 55 234)(24 285 56 233)(25 284 57 232)(26 283 58 231)(27 312 83 253)(28 311 84 252)(29 310 85 251)(30 309 86 250)(31 308 87 249)(32 307 88 248)(33 306 89 247)(34 305 90 246)(35 304 91 245)(36 303 92 244)(37 302 93 243)(38 301 94 242)(39 300 95 241)(40 299 96 240)(41 298 97 239)(42 297 98 238)(43 296 99 237)(44 295 100 236)(45 294 101 235)(46 293 102 260)(47 292 103 259)(48 291 104 258)(49 290 79 257)(50 289 80 256)(51 288 81 255)(52 287 82 254)(105 384 180 326)(106 383 181 325)(107 382 182 324)(108 381 157 323)(109 380 158 322)(110 379 159 321)(111 378 160 320)(112 377 161 319)(113 376 162 318)(114 375 163 317)(115 374 164 316)(116 373 165 315)(117 372 166 314)(118 371 167 313)(119 370 168 338)(120 369 169 337)(121 368 170 336)(122 367 171 335)(123 366 172 334)(124 365 173 333)(125 390 174 332)(126 389 175 331)(127 388 176 330)(128 387 177 329)(129 386 178 328)(130 385 179 327)(131 392 186 359)(132 391 187 358)(133 416 188 357)(134 415 189 356)(135 414 190 355)(136 413 191 354)(137 412 192 353)(138 411 193 352)(139 410 194 351)(140 409 195 350)(141 408 196 349)(142 407 197 348)(143 406 198 347)(144 405 199 346)(145 404 200 345)(146 403 201 344)(147 402 202 343)(148 401 203 342)(149 400 204 341)(150 399 205 340)(151 398 206 339)(152 397 207 364)(153 396 208 363)(154 395 183 362)(155 394 184 361)(156 393 185 360)
(1 137 30 129)(2 138 31 130)(3 139 32 105)(4 140 33 106)(5 141 34 107)(6 142 35 108)(7 143 36 109)(8 144 37 110)(9 145 38 111)(10 146 39 112)(11 147 40 113)(12 148 41 114)(13 149 42 115)(14 150 43 116)(15 151 44 117)(16 152 45 118)(17 153 46 119)(18 154 47 120)(19 155 48 121)(20 156 49 122)(21 131 50 123)(22 132 51 124)(23 133 52 125)(24 134 27 126)(25 135 28 127)(26 136 29 128)(53 186 80 172)(54 187 81 173)(55 188 82 174)(56 189 83 175)(57 190 84 176)(58 191 85 177)(59 192 86 178)(60 193 87 179)(61 194 88 180)(62 195 89 181)(63 196 90 182)(64 197 91 157)(65 198 92 158)(66 199 93 159)(67 200 94 160)(68 201 95 161)(69 202 96 162)(70 203 97 163)(71 204 98 164)(72 205 99 165)(73 206 100 166)(74 207 101 167)(75 208 102 168)(76 183 103 169)(77 184 104 170)(78 185 79 171)(209 345 255 320)(210 346 256 321)(211 347 257 322)(212 348 258 323)(213 349 259 324)(214 350 260 325)(215 351 235 326)(216 352 236 327)(217 353 237 328)(218 354 238 329)(219 355 239 330)(220 356 240 331)(221 357 241 332)(222 358 242 333)(223 359 243 334)(224 360 244 335)(225 361 245 336)(226 362 246 337)(227 363 247 338)(228 364 248 313)(229 339 249 314)(230 340 250 315)(231 341 251 316)(232 342 252 317)(233 343 253 318)(234 344 254 319)(261 404 288 378)(262 405 289 379)(263 406 290 380)(264 407 291 381)(265 408 292 382)(266 409 293 383)(267 410 294 384)(268 411 295 385)(269 412 296 386)(270 413 297 387)(271 414 298 388)(272 415 299 389)(273 416 300 390)(274 391 301 365)(275 392 302 366)(276 393 303 367)(277 394 304 368)(278 395 305 369)(279 396 306 370)(280 397 307 371)(281 398 308 372)(282 399 309 373)(283 400 310 374)(284 401 311 375)(285 402 312 376)(286 403 287 377)
G:=sub<Sym(416)| (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), (1,282,59,230)(2,281,60,229)(3,280,61,228)(4,279,62,227)(5,278,63,226)(6,277,64,225)(7,276,65,224)(8,275,66,223)(9,274,67,222)(10,273,68,221)(11,272,69,220)(12,271,70,219)(13,270,71,218)(14,269,72,217)(15,268,73,216)(16,267,74,215)(17,266,75,214)(18,265,76,213)(19,264,77,212)(20,263,78,211)(21,262,53,210)(22,261,54,209)(23,286,55,234)(24,285,56,233)(25,284,57,232)(26,283,58,231)(27,312,83,253)(28,311,84,252)(29,310,85,251)(30,309,86,250)(31,308,87,249)(32,307,88,248)(33,306,89,247)(34,305,90,246)(35,304,91,245)(36,303,92,244)(37,302,93,243)(38,301,94,242)(39,300,95,241)(40,299,96,240)(41,298,97,239)(42,297,98,238)(43,296,99,237)(44,295,100,236)(45,294,101,235)(46,293,102,260)(47,292,103,259)(48,291,104,258)(49,290,79,257)(50,289,80,256)(51,288,81,255)(52,287,82,254)(105,384,180,326)(106,383,181,325)(107,382,182,324)(108,381,157,323)(109,380,158,322)(110,379,159,321)(111,378,160,320)(112,377,161,319)(113,376,162,318)(114,375,163,317)(115,374,164,316)(116,373,165,315)(117,372,166,314)(118,371,167,313)(119,370,168,338)(120,369,169,337)(121,368,170,336)(122,367,171,335)(123,366,172,334)(124,365,173,333)(125,390,174,332)(126,389,175,331)(127,388,176,330)(128,387,177,329)(129,386,178,328)(130,385,179,327)(131,392,186,359)(132,391,187,358)(133,416,188,357)(134,415,189,356)(135,414,190,355)(136,413,191,354)(137,412,192,353)(138,411,193,352)(139,410,194,351)(140,409,195,350)(141,408,196,349)(142,407,197,348)(143,406,198,347)(144,405,199,346)(145,404,200,345)(146,403,201,344)(147,402,202,343)(148,401,203,342)(149,400,204,341)(150,399,205,340)(151,398,206,339)(152,397,207,364)(153,396,208,363)(154,395,183,362)(155,394,184,361)(156,393,185,360), (1,137,30,129)(2,138,31,130)(3,139,32,105)(4,140,33,106)(5,141,34,107)(6,142,35,108)(7,143,36,109)(8,144,37,110)(9,145,38,111)(10,146,39,112)(11,147,40,113)(12,148,41,114)(13,149,42,115)(14,150,43,116)(15,151,44,117)(16,152,45,118)(17,153,46,119)(18,154,47,120)(19,155,48,121)(20,156,49,122)(21,131,50,123)(22,132,51,124)(23,133,52,125)(24,134,27,126)(25,135,28,127)(26,136,29,128)(53,186,80,172)(54,187,81,173)(55,188,82,174)(56,189,83,175)(57,190,84,176)(58,191,85,177)(59,192,86,178)(60,193,87,179)(61,194,88,180)(62,195,89,181)(63,196,90,182)(64,197,91,157)(65,198,92,158)(66,199,93,159)(67,200,94,160)(68,201,95,161)(69,202,96,162)(70,203,97,163)(71,204,98,164)(72,205,99,165)(73,206,100,166)(74,207,101,167)(75,208,102,168)(76,183,103,169)(77,184,104,170)(78,185,79,171)(209,345,255,320)(210,346,256,321)(211,347,257,322)(212,348,258,323)(213,349,259,324)(214,350,260,325)(215,351,235,326)(216,352,236,327)(217,353,237,328)(218,354,238,329)(219,355,239,330)(220,356,240,331)(221,357,241,332)(222,358,242,333)(223,359,243,334)(224,360,244,335)(225,361,245,336)(226,362,246,337)(227,363,247,338)(228,364,248,313)(229,339,249,314)(230,340,250,315)(231,341,251,316)(232,342,252,317)(233,343,253,318)(234,344,254,319)(261,404,288,378)(262,405,289,379)(263,406,290,380)(264,407,291,381)(265,408,292,382)(266,409,293,383)(267,410,294,384)(268,411,295,385)(269,412,296,386)(270,413,297,387)(271,414,298,388)(272,415,299,389)(273,416,300,390)(274,391,301,365)(275,392,302,366)(276,393,303,367)(277,394,304,368)(278,395,305,369)(279,396,306,370)(280,397,307,371)(281,398,308,372)(282,399,309,373)(283,400,310,374)(284,401,311,375)(285,402,312,376)(286,403,287,377)>;
G:=Group( (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), (1,282,59,230)(2,281,60,229)(3,280,61,228)(4,279,62,227)(5,278,63,226)(6,277,64,225)(7,276,65,224)(8,275,66,223)(9,274,67,222)(10,273,68,221)(11,272,69,220)(12,271,70,219)(13,270,71,218)(14,269,72,217)(15,268,73,216)(16,267,74,215)(17,266,75,214)(18,265,76,213)(19,264,77,212)(20,263,78,211)(21,262,53,210)(22,261,54,209)(23,286,55,234)(24,285,56,233)(25,284,57,232)(26,283,58,231)(27,312,83,253)(28,311,84,252)(29,310,85,251)(30,309,86,250)(31,308,87,249)(32,307,88,248)(33,306,89,247)(34,305,90,246)(35,304,91,245)(36,303,92,244)(37,302,93,243)(38,301,94,242)(39,300,95,241)(40,299,96,240)(41,298,97,239)(42,297,98,238)(43,296,99,237)(44,295,100,236)(45,294,101,235)(46,293,102,260)(47,292,103,259)(48,291,104,258)(49,290,79,257)(50,289,80,256)(51,288,81,255)(52,287,82,254)(105,384,180,326)(106,383,181,325)(107,382,182,324)(108,381,157,323)(109,380,158,322)(110,379,159,321)(111,378,160,320)(112,377,161,319)(113,376,162,318)(114,375,163,317)(115,374,164,316)(116,373,165,315)(117,372,166,314)(118,371,167,313)(119,370,168,338)(120,369,169,337)(121,368,170,336)(122,367,171,335)(123,366,172,334)(124,365,173,333)(125,390,174,332)(126,389,175,331)(127,388,176,330)(128,387,177,329)(129,386,178,328)(130,385,179,327)(131,392,186,359)(132,391,187,358)(133,416,188,357)(134,415,189,356)(135,414,190,355)(136,413,191,354)(137,412,192,353)(138,411,193,352)(139,410,194,351)(140,409,195,350)(141,408,196,349)(142,407,197,348)(143,406,198,347)(144,405,199,346)(145,404,200,345)(146,403,201,344)(147,402,202,343)(148,401,203,342)(149,400,204,341)(150,399,205,340)(151,398,206,339)(152,397,207,364)(153,396,208,363)(154,395,183,362)(155,394,184,361)(156,393,185,360), (1,137,30,129)(2,138,31,130)(3,139,32,105)(4,140,33,106)(5,141,34,107)(6,142,35,108)(7,143,36,109)(8,144,37,110)(9,145,38,111)(10,146,39,112)(11,147,40,113)(12,148,41,114)(13,149,42,115)(14,150,43,116)(15,151,44,117)(16,152,45,118)(17,153,46,119)(18,154,47,120)(19,155,48,121)(20,156,49,122)(21,131,50,123)(22,132,51,124)(23,133,52,125)(24,134,27,126)(25,135,28,127)(26,136,29,128)(53,186,80,172)(54,187,81,173)(55,188,82,174)(56,189,83,175)(57,190,84,176)(58,191,85,177)(59,192,86,178)(60,193,87,179)(61,194,88,180)(62,195,89,181)(63,196,90,182)(64,197,91,157)(65,198,92,158)(66,199,93,159)(67,200,94,160)(68,201,95,161)(69,202,96,162)(70,203,97,163)(71,204,98,164)(72,205,99,165)(73,206,100,166)(74,207,101,167)(75,208,102,168)(76,183,103,169)(77,184,104,170)(78,185,79,171)(209,345,255,320)(210,346,256,321)(211,347,257,322)(212,348,258,323)(213,349,259,324)(214,350,260,325)(215,351,235,326)(216,352,236,327)(217,353,237,328)(218,354,238,329)(219,355,239,330)(220,356,240,331)(221,357,241,332)(222,358,242,333)(223,359,243,334)(224,360,244,335)(225,361,245,336)(226,362,246,337)(227,363,247,338)(228,364,248,313)(229,339,249,314)(230,340,250,315)(231,341,251,316)(232,342,252,317)(233,343,253,318)(234,344,254,319)(261,404,288,378)(262,405,289,379)(263,406,290,380)(264,407,291,381)(265,408,292,382)(266,409,293,383)(267,410,294,384)(268,411,295,385)(269,412,296,386)(270,413,297,387)(271,414,298,388)(272,415,299,389)(273,416,300,390)(274,391,301,365)(275,392,302,366)(276,393,303,367)(277,394,304,368)(278,395,305,369)(279,396,306,370)(280,397,307,371)(281,398,308,372)(282,399,309,373)(283,400,310,374)(284,401,311,375)(285,402,312,376)(286,403,287,377) );
G=PermutationGroup([[(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)], [(1,282,59,230),(2,281,60,229),(3,280,61,228),(4,279,62,227),(5,278,63,226),(6,277,64,225),(7,276,65,224),(8,275,66,223),(9,274,67,222),(10,273,68,221),(11,272,69,220),(12,271,70,219),(13,270,71,218),(14,269,72,217),(15,268,73,216),(16,267,74,215),(17,266,75,214),(18,265,76,213),(19,264,77,212),(20,263,78,211),(21,262,53,210),(22,261,54,209),(23,286,55,234),(24,285,56,233),(25,284,57,232),(26,283,58,231),(27,312,83,253),(28,311,84,252),(29,310,85,251),(30,309,86,250),(31,308,87,249),(32,307,88,248),(33,306,89,247),(34,305,90,246),(35,304,91,245),(36,303,92,244),(37,302,93,243),(38,301,94,242),(39,300,95,241),(40,299,96,240),(41,298,97,239),(42,297,98,238),(43,296,99,237),(44,295,100,236),(45,294,101,235),(46,293,102,260),(47,292,103,259),(48,291,104,258),(49,290,79,257),(50,289,80,256),(51,288,81,255),(52,287,82,254),(105,384,180,326),(106,383,181,325),(107,382,182,324),(108,381,157,323),(109,380,158,322),(110,379,159,321),(111,378,160,320),(112,377,161,319),(113,376,162,318),(114,375,163,317),(115,374,164,316),(116,373,165,315),(117,372,166,314),(118,371,167,313),(119,370,168,338),(120,369,169,337),(121,368,170,336),(122,367,171,335),(123,366,172,334),(124,365,173,333),(125,390,174,332),(126,389,175,331),(127,388,176,330),(128,387,177,329),(129,386,178,328),(130,385,179,327),(131,392,186,359),(132,391,187,358),(133,416,188,357),(134,415,189,356),(135,414,190,355),(136,413,191,354),(137,412,192,353),(138,411,193,352),(139,410,194,351),(140,409,195,350),(141,408,196,349),(142,407,197,348),(143,406,198,347),(144,405,199,346),(145,404,200,345),(146,403,201,344),(147,402,202,343),(148,401,203,342),(149,400,204,341),(150,399,205,340),(151,398,206,339),(152,397,207,364),(153,396,208,363),(154,395,183,362),(155,394,184,361),(156,393,185,360)], [(1,137,30,129),(2,138,31,130),(3,139,32,105),(4,140,33,106),(5,141,34,107),(6,142,35,108),(7,143,36,109),(8,144,37,110),(9,145,38,111),(10,146,39,112),(11,147,40,113),(12,148,41,114),(13,149,42,115),(14,150,43,116),(15,151,44,117),(16,152,45,118),(17,153,46,119),(18,154,47,120),(19,155,48,121),(20,156,49,122),(21,131,50,123),(22,132,51,124),(23,133,52,125),(24,134,27,126),(25,135,28,127),(26,136,29,128),(53,186,80,172),(54,187,81,173),(55,188,82,174),(56,189,83,175),(57,190,84,176),(58,191,85,177),(59,192,86,178),(60,193,87,179),(61,194,88,180),(62,195,89,181),(63,196,90,182),(64,197,91,157),(65,198,92,158),(66,199,93,159),(67,200,94,160),(68,201,95,161),(69,202,96,162),(70,203,97,163),(71,204,98,164),(72,205,99,165),(73,206,100,166),(74,207,101,167),(75,208,102,168),(76,183,103,169),(77,184,104,170),(78,185,79,171),(209,345,255,320),(210,346,256,321),(211,347,257,322),(212,348,258,323),(213,349,259,324),(214,350,260,325),(215,351,235,326),(216,352,236,327),(217,353,237,328),(218,354,238,329),(219,355,239,330),(220,356,240,331),(221,357,241,332),(222,358,242,333),(223,359,243,334),(224,360,244,335),(225,361,245,336),(226,362,246,337),(227,363,247,338),(228,364,248,313),(229,339,249,314),(230,340,250,315),(231,341,251,316),(232,342,252,317),(233,343,253,318),(234,344,254,319),(261,404,288,378),(262,405,289,379),(263,406,290,380),(264,407,291,381),(265,408,292,382),(266,409,293,383),(267,410,294,384),(268,411,295,385),(269,412,296,386),(270,413,297,387),(271,414,298,388),(272,415,299,389),(273,416,300,390),(274,391,301,365),(275,392,302,366),(276,393,303,367),(277,394,304,368),(278,395,305,369),(279,396,306,370),(280,397,307,371),(281,398,308,372),(282,399,309,373),(283,400,310,374),(284,401,311,375),(285,402,312,376),(286,403,287,377)]])
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 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | Q8 | D13 | Dic13 | D26 | Dic26 | C4×D13 | D52 | C13⋊D4 |
| kernel | C26.10C42 | C22×Dic13 | C22×C52 | C2×Dic13 | C2×C52 | C2×C26 | C2×C26 | C22×C4 | C2×C4 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 8 | 4 | 3 | 1 | 6 | 12 | 6 | 12 | 24 | 12 | 24 |
Matrix representation of C26.10C42 ►in GL6(𝔽53)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 49 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 0 | 0 | 52 |
| 52 | 0 | 0 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 28 |
| 0 | 0 | 0 | 0 | 23 | 24 |
| 23 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 0 | 0 | 0 |
| 0 | 0 | 0 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 44 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(53))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,49,0,0,0,0,0,0,52,0,0,0,0,0,0,52],[52,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,29,23,0,0,0,0,28,24],[23,0,0,0,0,0,0,1,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,0,52,0,0,0,0,0,44,1] >;
C26.10C42 in GAP, Magma, Sage, TeX
C_{26}._{10}C_4^2 % in TeX
G:=Group("C26.10C4^2"); // GroupNames label
G:=SmallGroup(416,38);
// by ID
G=gap.SmallGroup(416,38);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,55,13829]);
// Polycyclic
G:=Group<a,b,c|a^26=b^4=c^4=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^13*b>;
// generators/relations