metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C26.4C42, C26.2M4(2), C13⋊C8⋊2C4, (C2×C52).2C4, C13⋊2(C8⋊C4), (C4×Dic13).7C2, C2.2(C52.C4), Dic13.10(C2×C4), (C2×Dic13).10C4, C2.1(C13⋊M4(2)), (C2×Dic13).50C22, C2.4(C4×C13⋊C4), (C2×C13⋊C8).2C2, (C2×C4).2(C13⋊C4), (C2×C26).5(C2×C4), C22.10(C2×C13⋊C4), SmallGroup(416,77)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C26.C42
G = < a,b,c | a26=c4=1, b4=a13, bab-1=a5, ac=ca, cbc-1=a13b >
Smallest permutation representation of C26.C42
►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)
(1 270 55 218 14 283 68 231)(2 265 54 223 15 278 67 210)(3 286 53 228 16 273 66 215)(4 281 78 233 17 268 65 220)(5 276 77 212 18 263 64 225)(6 271 76 217 19 284 63 230)(7 266 75 222 20 279 62 209)(8 261 74 227 21 274 61 214)(9 282 73 232 22 269 60 219)(10 277 72 211 23 264 59 224)(11 272 71 216 24 285 58 229)(12 267 70 221 25 280 57 234)(13 262 69 226 26 275 56 213)(27 296 85 235 40 309 98 248)(28 291 84 240 41 304 97 253)(29 312 83 245 42 299 96 258)(30 307 82 250 43 294 95 237)(31 302 81 255 44 289 94 242)(32 297 80 260 45 310 93 247)(33 292 79 239 46 305 92 252)(34 287 104 244 47 300 91 257)(35 308 103 249 48 295 90 236)(36 303 102 254 49 290 89 241)(37 298 101 259 50 311 88 246)(38 293 100 238 51 306 87 251)(39 288 99 243 52 301 86 256)(105 386 174 328 118 373 161 315)(106 381 173 333 119 368 160 320)(107 376 172 338 120 389 159 325)(108 371 171 317 121 384 158 330)(109 366 170 322 122 379 157 335)(110 387 169 327 123 374 182 314)(111 382 168 332 124 369 181 319)(112 377 167 337 125 390 180 324)(113 372 166 316 126 385 179 329)(114 367 165 321 127 380 178 334)(115 388 164 326 128 375 177 313)(116 383 163 331 129 370 176 318)(117 378 162 336 130 365 175 323)(131 411 198 355 144 398 185 342)(132 406 197 360 145 393 184 347)(133 401 196 339 146 414 183 352)(134 396 195 344 147 409 208 357)(135 391 194 349 148 404 207 362)(136 412 193 354 149 399 206 341)(137 407 192 359 150 394 205 346)(138 402 191 364 151 415 204 351)(139 397 190 343 152 410 203 356)(140 392 189 348 153 405 202 361)(141 413 188 353 154 400 201 340)(142 408 187 358 155 395 200 345)(143 403 186 363 156 416 199 350)
(1 140 28 109)(2 141 29 110)(3 142 30 111)(4 143 31 112)(5 144 32 113)(6 145 33 114)(7 146 34 115)(8 147 35 116)(9 148 36 117)(10 149 37 118)(11 150 38 119)(12 151 39 120)(13 152 40 121)(14 153 41 122)(15 154 42 123)(16 155 43 124)(17 156 44 125)(18 131 45 126)(19 132 46 127)(20 133 47 128)(21 134 48 129)(22 135 49 130)(23 136 50 105)(24 137 51 106)(25 138 52 107)(26 139 27 108)(53 187 82 168)(54 188 83 169)(55 189 84 170)(56 190 85 171)(57 191 86 172)(58 192 87 173)(59 193 88 174)(60 194 89 175)(61 195 90 176)(62 196 91 177)(63 197 92 178)(64 198 93 179)(65 199 94 180)(66 200 95 181)(67 201 96 182)(68 202 97 157)(69 203 98 158)(70 204 99 159)(71 205 100 160)(72 206 101 161)(73 207 102 162)(74 208 103 163)(75 183 104 164)(76 184 79 165)(77 185 80 166)(78 186 81 167)(209 352 257 326)(210 353 258 327)(211 354 259 328)(212 355 260 329)(213 356 235 330)(214 357 236 331)(215 358 237 332)(216 359 238 333)(217 360 239 334)(218 361 240 335)(219 362 241 336)(220 363 242 337)(221 364 243 338)(222 339 244 313)(223 340 245 314)(224 341 246 315)(225 342 247 316)(226 343 248 317)(227 344 249 318)(228 345 250 319)(229 346 251 320)(230 347 252 321)(231 348 253 322)(232 349 254 323)(233 350 255 324)(234 351 256 325)(261 396 308 370)(262 397 309 371)(263 398 310 372)(264 399 311 373)(265 400 312 374)(266 401 287 375)(267 402 288 376)(268 403 289 377)(269 404 290 378)(270 405 291 379)(271 406 292 380)(272 407 293 381)(273 408 294 382)(274 409 295 383)(275 410 296 384)(276 411 297 385)(277 412 298 386)(278 413 299 387)(279 414 300 388)(280 415 301 389)(281 416 302 390)(282 391 303 365)(283 392 304 366)(284 393 305 367)(285 394 306 368)(286 395 307 369)
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), (1,270,55,218,14,283,68,231)(2,265,54,223,15,278,67,210)(3,286,53,228,16,273,66,215)(4,281,78,233,17,268,65,220)(5,276,77,212,18,263,64,225)(6,271,76,217,19,284,63,230)(7,266,75,222,20,279,62,209)(8,261,74,227,21,274,61,214)(9,282,73,232,22,269,60,219)(10,277,72,211,23,264,59,224)(11,272,71,216,24,285,58,229)(12,267,70,221,25,280,57,234)(13,262,69,226,26,275,56,213)(27,296,85,235,40,309,98,248)(28,291,84,240,41,304,97,253)(29,312,83,245,42,299,96,258)(30,307,82,250,43,294,95,237)(31,302,81,255,44,289,94,242)(32,297,80,260,45,310,93,247)(33,292,79,239,46,305,92,252)(34,287,104,244,47,300,91,257)(35,308,103,249,48,295,90,236)(36,303,102,254,49,290,89,241)(37,298,101,259,50,311,88,246)(38,293,100,238,51,306,87,251)(39,288,99,243,52,301,86,256)(105,386,174,328,118,373,161,315)(106,381,173,333,119,368,160,320)(107,376,172,338,120,389,159,325)(108,371,171,317,121,384,158,330)(109,366,170,322,122,379,157,335)(110,387,169,327,123,374,182,314)(111,382,168,332,124,369,181,319)(112,377,167,337,125,390,180,324)(113,372,166,316,126,385,179,329)(114,367,165,321,127,380,178,334)(115,388,164,326,128,375,177,313)(116,383,163,331,129,370,176,318)(117,378,162,336,130,365,175,323)(131,411,198,355,144,398,185,342)(132,406,197,360,145,393,184,347)(133,401,196,339,146,414,183,352)(134,396,195,344,147,409,208,357)(135,391,194,349,148,404,207,362)(136,412,193,354,149,399,206,341)(137,407,192,359,150,394,205,346)(138,402,191,364,151,415,204,351)(139,397,190,343,152,410,203,356)(140,392,189,348,153,405,202,361)(141,413,188,353,154,400,201,340)(142,408,187,358,155,395,200,345)(143,403,186,363,156,416,199,350), (1,140,28,109)(2,141,29,110)(3,142,30,111)(4,143,31,112)(5,144,32,113)(6,145,33,114)(7,146,34,115)(8,147,35,116)(9,148,36,117)(10,149,37,118)(11,150,38,119)(12,151,39,120)(13,152,40,121)(14,153,41,122)(15,154,42,123)(16,155,43,124)(17,156,44,125)(18,131,45,126)(19,132,46,127)(20,133,47,128)(21,134,48,129)(22,135,49,130)(23,136,50,105)(24,137,51,106)(25,138,52,107)(26,139,27,108)(53,187,82,168)(54,188,83,169)(55,189,84,170)(56,190,85,171)(57,191,86,172)(58,192,87,173)(59,193,88,174)(60,194,89,175)(61,195,90,176)(62,196,91,177)(63,197,92,178)(64,198,93,179)(65,199,94,180)(66,200,95,181)(67,201,96,182)(68,202,97,157)(69,203,98,158)(70,204,99,159)(71,205,100,160)(72,206,101,161)(73,207,102,162)(74,208,103,163)(75,183,104,164)(76,184,79,165)(77,185,80,166)(78,186,81,167)(209,352,257,326)(210,353,258,327)(211,354,259,328)(212,355,260,329)(213,356,235,330)(214,357,236,331)(215,358,237,332)(216,359,238,333)(217,360,239,334)(218,361,240,335)(219,362,241,336)(220,363,242,337)(221,364,243,338)(222,339,244,313)(223,340,245,314)(224,341,246,315)(225,342,247,316)(226,343,248,317)(227,344,249,318)(228,345,250,319)(229,346,251,320)(230,347,252,321)(231,348,253,322)(232,349,254,323)(233,350,255,324)(234,351,256,325)(261,396,308,370)(262,397,309,371)(263,398,310,372)(264,399,311,373)(265,400,312,374)(266,401,287,375)(267,402,288,376)(268,403,289,377)(269,404,290,378)(270,405,291,379)(271,406,292,380)(272,407,293,381)(273,408,294,382)(274,409,295,383)(275,410,296,384)(276,411,297,385)(277,412,298,386)(278,413,299,387)(279,414,300,388)(280,415,301,389)(281,416,302,390)(282,391,303,365)(283,392,304,366)(284,393,305,367)(285,394,306,368)(286,395,307,369)>;
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), (1,270,55,218,14,283,68,231)(2,265,54,223,15,278,67,210)(3,286,53,228,16,273,66,215)(4,281,78,233,17,268,65,220)(5,276,77,212,18,263,64,225)(6,271,76,217,19,284,63,230)(7,266,75,222,20,279,62,209)(8,261,74,227,21,274,61,214)(9,282,73,232,22,269,60,219)(10,277,72,211,23,264,59,224)(11,272,71,216,24,285,58,229)(12,267,70,221,25,280,57,234)(13,262,69,226,26,275,56,213)(27,296,85,235,40,309,98,248)(28,291,84,240,41,304,97,253)(29,312,83,245,42,299,96,258)(30,307,82,250,43,294,95,237)(31,302,81,255,44,289,94,242)(32,297,80,260,45,310,93,247)(33,292,79,239,46,305,92,252)(34,287,104,244,47,300,91,257)(35,308,103,249,48,295,90,236)(36,303,102,254,49,290,89,241)(37,298,101,259,50,311,88,246)(38,293,100,238,51,306,87,251)(39,288,99,243,52,301,86,256)(105,386,174,328,118,373,161,315)(106,381,173,333,119,368,160,320)(107,376,172,338,120,389,159,325)(108,371,171,317,121,384,158,330)(109,366,170,322,122,379,157,335)(110,387,169,327,123,374,182,314)(111,382,168,332,124,369,181,319)(112,377,167,337,125,390,180,324)(113,372,166,316,126,385,179,329)(114,367,165,321,127,380,178,334)(115,388,164,326,128,375,177,313)(116,383,163,331,129,370,176,318)(117,378,162,336,130,365,175,323)(131,411,198,355,144,398,185,342)(132,406,197,360,145,393,184,347)(133,401,196,339,146,414,183,352)(134,396,195,344,147,409,208,357)(135,391,194,349,148,404,207,362)(136,412,193,354,149,399,206,341)(137,407,192,359,150,394,205,346)(138,402,191,364,151,415,204,351)(139,397,190,343,152,410,203,356)(140,392,189,348,153,405,202,361)(141,413,188,353,154,400,201,340)(142,408,187,358,155,395,200,345)(143,403,186,363,156,416,199,350), (1,140,28,109)(2,141,29,110)(3,142,30,111)(4,143,31,112)(5,144,32,113)(6,145,33,114)(7,146,34,115)(8,147,35,116)(9,148,36,117)(10,149,37,118)(11,150,38,119)(12,151,39,120)(13,152,40,121)(14,153,41,122)(15,154,42,123)(16,155,43,124)(17,156,44,125)(18,131,45,126)(19,132,46,127)(20,133,47,128)(21,134,48,129)(22,135,49,130)(23,136,50,105)(24,137,51,106)(25,138,52,107)(26,139,27,108)(53,187,82,168)(54,188,83,169)(55,189,84,170)(56,190,85,171)(57,191,86,172)(58,192,87,173)(59,193,88,174)(60,194,89,175)(61,195,90,176)(62,196,91,177)(63,197,92,178)(64,198,93,179)(65,199,94,180)(66,200,95,181)(67,201,96,182)(68,202,97,157)(69,203,98,158)(70,204,99,159)(71,205,100,160)(72,206,101,161)(73,207,102,162)(74,208,103,163)(75,183,104,164)(76,184,79,165)(77,185,80,166)(78,186,81,167)(209,352,257,326)(210,353,258,327)(211,354,259,328)(212,355,260,329)(213,356,235,330)(214,357,236,331)(215,358,237,332)(216,359,238,333)(217,360,239,334)(218,361,240,335)(219,362,241,336)(220,363,242,337)(221,364,243,338)(222,339,244,313)(223,340,245,314)(224,341,246,315)(225,342,247,316)(226,343,248,317)(227,344,249,318)(228,345,250,319)(229,346,251,320)(230,347,252,321)(231,348,253,322)(232,349,254,323)(233,350,255,324)(234,351,256,325)(261,396,308,370)(262,397,309,371)(263,398,310,372)(264,399,311,373)(265,400,312,374)(266,401,287,375)(267,402,288,376)(268,403,289,377)(269,404,290,378)(270,405,291,379)(271,406,292,380)(272,407,293,381)(273,408,294,382)(274,409,295,383)(275,410,296,384)(276,411,297,385)(277,412,298,386)(278,413,299,387)(279,414,300,388)(280,415,301,389)(281,416,302,390)(282,391,303,365)(283,392,304,366)(284,393,305,367)(285,394,306,368)(286,395,307,369) );
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)], [(1,270,55,218,14,283,68,231),(2,265,54,223,15,278,67,210),(3,286,53,228,16,273,66,215),(4,281,78,233,17,268,65,220),(5,276,77,212,18,263,64,225),(6,271,76,217,19,284,63,230),(7,266,75,222,20,279,62,209),(8,261,74,227,21,274,61,214),(9,282,73,232,22,269,60,219),(10,277,72,211,23,264,59,224),(11,272,71,216,24,285,58,229),(12,267,70,221,25,280,57,234),(13,262,69,226,26,275,56,213),(27,296,85,235,40,309,98,248),(28,291,84,240,41,304,97,253),(29,312,83,245,42,299,96,258),(30,307,82,250,43,294,95,237),(31,302,81,255,44,289,94,242),(32,297,80,260,45,310,93,247),(33,292,79,239,46,305,92,252),(34,287,104,244,47,300,91,257),(35,308,103,249,48,295,90,236),(36,303,102,254,49,290,89,241),(37,298,101,259,50,311,88,246),(38,293,100,238,51,306,87,251),(39,288,99,243,52,301,86,256),(105,386,174,328,118,373,161,315),(106,381,173,333,119,368,160,320),(107,376,172,338,120,389,159,325),(108,371,171,317,121,384,158,330),(109,366,170,322,122,379,157,335),(110,387,169,327,123,374,182,314),(111,382,168,332,124,369,181,319),(112,377,167,337,125,390,180,324),(113,372,166,316,126,385,179,329),(114,367,165,321,127,380,178,334),(115,388,164,326,128,375,177,313),(116,383,163,331,129,370,176,318),(117,378,162,336,130,365,175,323),(131,411,198,355,144,398,185,342),(132,406,197,360,145,393,184,347),(133,401,196,339,146,414,183,352),(134,396,195,344,147,409,208,357),(135,391,194,349,148,404,207,362),(136,412,193,354,149,399,206,341),(137,407,192,359,150,394,205,346),(138,402,191,364,151,415,204,351),(139,397,190,343,152,410,203,356),(140,392,189,348,153,405,202,361),(141,413,188,353,154,400,201,340),(142,408,187,358,155,395,200,345),(143,403,186,363,156,416,199,350)], [(1,140,28,109),(2,141,29,110),(3,142,30,111),(4,143,31,112),(5,144,32,113),(6,145,33,114),(7,146,34,115),(8,147,35,116),(9,148,36,117),(10,149,37,118),(11,150,38,119),(12,151,39,120),(13,152,40,121),(14,153,41,122),(15,154,42,123),(16,155,43,124),(17,156,44,125),(18,131,45,126),(19,132,46,127),(20,133,47,128),(21,134,48,129),(22,135,49,130),(23,136,50,105),(24,137,51,106),(25,138,52,107),(26,139,27,108),(53,187,82,168),(54,188,83,169),(55,189,84,170),(56,190,85,171),(57,191,86,172),(58,192,87,173),(59,193,88,174),(60,194,89,175),(61,195,90,176),(62,196,91,177),(63,197,92,178),(64,198,93,179),(65,199,94,180),(66,200,95,181),(67,201,96,182),(68,202,97,157),(69,203,98,158),(70,204,99,159),(71,205,100,160),(72,206,101,161),(73,207,102,162),(74,208,103,163),(75,183,104,164),(76,184,79,165),(77,185,80,166),(78,186,81,167),(209,352,257,326),(210,353,258,327),(211,354,259,328),(212,355,260,329),(213,356,235,330),(214,357,236,331),(215,358,237,332),(216,359,238,333),(217,360,239,334),(218,361,240,335),(219,362,241,336),(220,363,242,337),(221,364,243,338),(222,339,244,313),(223,340,245,314),(224,341,246,315),(225,342,247,316),(226,343,248,317),(227,344,249,318),(228,345,250,319),(229,346,251,320),(230,347,252,321),(231,348,253,322),(232,349,254,323),(233,350,255,324),(234,351,256,325),(261,396,308,370),(262,397,309,371),(263,398,310,372),(264,399,311,373),(265,400,312,374),(266,401,287,375),(267,402,288,376),(268,403,289,377),(269,404,290,378),(270,405,291,379),(271,406,292,380),(272,407,293,381),(273,408,294,382),(274,409,295,383),(275,410,296,384),(276,411,297,385),(277,412,298,386),(278,413,299,387),(279,414,300,388),(280,415,301,389),(281,416,302,390),(282,391,303,365),(283,392,304,366),(284,393,305,367),(285,394,306,368),(286,395,307,369)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 13A | 13B | 13C | 26A | ··· | 26I | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 13 | 13 | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 13 | 13 | 13 | 13 | 26 | 26 | 26 | ··· | 26 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | M4(2) | C13⋊C4 | C2×C13⋊C4 | C52.C4 | C4×C13⋊C4 | C13⋊M4(2) |
| kernel | C26.C42 | C4×Dic13 | C2×C13⋊C8 | C13⋊C8 | C2×Dic13 | C2×C52 | C26 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 4 | 3 | 3 | 6 | 6 | 6 |
Matrix representation of C26.C42 ►in GL6(𝔽313)
| 312 | 0 | 0 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 273 | 281 | 241 |
| 0 | 0 | 40 | 273 | 282 | 0 |
| 0 | 0 | 72 | 0 | 241 | 210 |
| 0 | 0 | 30 | 282 | 31 | 283 |
| 43 | 48 | 0 | 0 | 0 | 0 |
| 275 | 270 | 0 | 0 | 0 | 0 |
| 0 | 0 | 296 | 215 | 158 | 103 |
| 0 | 0 | 213 | 279 | 179 | 27 |
| 0 | 0 | 299 | 275 | 301 | 218 |
| 0 | 0 | 83 | 42 | 235 | 63 |
| 312 | 249 | 0 | 0 | 0 | 0 |
| 225 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 278 | 272 | 289 | 176 |
| 0 | 0 | 178 | 59 | 113 | 176 |
| 0 | 0 | 41 | 272 | 213 | 0 |
| 0 | 0 | 65 | 0 | 248 | 76 |
G:=sub<GL(6,GF(313))| [312,0,0,0,0,0,0,312,0,0,0,0,0,0,41,40,72,30,0,0,273,273,0,282,0,0,281,282,241,31,0,0,241,0,210,283],[43,275,0,0,0,0,48,270,0,0,0,0,0,0,296,213,299,83,0,0,215,279,275,42,0,0,158,179,301,235,0,0,103,27,218,63],[312,225,0,0,0,0,249,1,0,0,0,0,0,0,278,178,41,65,0,0,272,59,272,0,0,0,289,113,213,248,0,0,176,176,0,76] >;
C26.C42 in GAP, Magma, Sage, TeX
C_{26}.C_4^2 % in TeX
G:=Group("C26.C4^2"); // GroupNames label
G:=SmallGroup(416,77);
// by ID
G=gap.SmallGroup(416,77);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,217,55,86,9221,3473]);
// Polycyclic
G:=Group<a,b,c|a^26=c^4=1,b^4=a^13,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^13*b>;
// generators/relations
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