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G = C26.C42order 416 = 25·13

4th non-split extension by C26 of C42 acting via C42/C4=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C26.4C42, C26.2M4(2), C13⋊C82C4, (C2×C52).2C4, C132(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

C1C26 — C26.C42
C1C13C26Dic13C2×Dic13C2×C13⋊C8 — C26.C42
C13C26 — C26.C42
C1C22C2×C4

Generators and relations for C26.C42
 G = < a,b,c | a26=c4=1, b4=a13, bab-1=a5, ac=ca, cbc-1=a13b >

2C4
13C4
13C4
26C4
13C8
13C8
13C8
13C8
13C2×C4
13C2×C4
2Dic13
2C52
13C2×C8
13C42
13C2×C8
13C8⋊C4

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

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H8A···8H13A13B13C26A···26I52A···52L
order1222444444448···813131326···2652···52
size11112213131313262626···264444···44···4

44 irreducible representations

dim111111244444
type+++++-
imageC1C2C2C4C4C4M4(2)C13⋊C4C2×C13⋊C4C52.C4C4×C13⋊C4C13⋊M4(2)
kernelC26.C42C4×Dic13C2×C13⋊C8C13⋊C8C2×Dic13C2×C52C26C2×C4C22C2C2C2
# reps112822433666

Matrix representation of C26.C42 in GL6(𝔽313)

31200000
03120000
0041273281241
00402732820
00720241210
003028231283
,
43480000
2752700000
00296215158103
0021327917927
00299275301218
00834223563
,
3122490000
22510000
00278272289176
0017859113176
00412722130
0065024876

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

Export

Subgroup lattice of C26.C42 in TeX

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