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G = C26.10C42order 416 = 25·13

5th non-split extension by C26 of C42 acting via C42/C2×C4=C2

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(C523C4), (C22×C4).2D13, C22.12(C4×D13), C2.2(D26⋊C4), C26.22(C22⋊C4), C132(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

C1C26 — C26.10C42
C1C13C26C2×C26C22×C26C22×Dic13 — C26.10C42
C13C26 — C26.10C42
C1C23C22×C4

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 [×3], C2 [×4], C4 [×6], C22 [×3], C22 [×4], C2×C4 [×2], C2×C4 [×10], C23, C13, C22×C4, C22×C4 [×2], C26 [×3], C26 [×4], C2.C42, Dic13 [×4], C52 [×2], C2×C26 [×3], C2×C26 [×4], C2×Dic13 [×4], C2×Dic13 [×4], C2×C52 [×2], C2×C52 [×2], C22×C26, C22×Dic13 [×2], C22×C52, C26.10C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D13, C2.C42, Dic13 [×2], D26, Dic26, C4×D13 [×2], D52, C2×Dic13, C13⋊D4 [×2], C4×Dic13, C26.D4 [×2], C523C4, D26⋊C4 [×2], C23.D13, C26.10C42

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

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

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

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

116 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4L13A···13F26A···26AP52A···52AV
order12···244444···413···1326···2652···52
size11···1222226···262···22···22···2

116 irreducible representations

dim11111222222222
type++++-+-+-+
imageC1C2C2C4C4D4Q8D13Dic13D26Dic26C4×D13D52C13⋊D4
kernelC26.10C42C22×Dic13C22×C52C2×Dic13C2×C52C2×C26C2×C26C22×C4C2×C4C23C22C22C22C22
# reps1218431612612241224

Matrix representation of C26.10C42 in GL6(𝔽53)

100000
010000
0013000
0004900
0000520
0000052
,
5200000
0300000
000100
001000
00002928
00002324
,
2300000
010000
0052000
0005200
00005244
000001

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

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