metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C52.2D4, C4.10D52, C26.4Q16, Dic26⋊6C4, C26.5SD16, C4⋊C4.3D13, C4.2(C4×D13), C52.25(C2×C4), (C2×C26).31D4, (C2×C4).36D26, C13⋊2(Q8⋊C4), C2.2(D4.D13), (C2×C52).11C22, (C2×Dic26).6C2, C2.2(C13⋊Q16), C2.6(D26⋊C4), C26.15(C22⋊C4), C22.15(C13⋊D4), (C13×C4⋊C4).3C2, (C2×C13⋊2C8).3C2, SmallGroup(416,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C26.Q16
G = < a,b,c | a26=b8=1, c2=a13b4, bab-1=a-1, ac=ca, cbc-1=a13b-1 >
Smallest permutation representation of C26.Q16
►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 28 363 280 288 105 375 327)(2 27 364 279 289 130 376 326)(3 52 339 278 290 129 377 325)(4 51 340 277 291 128 378 324)(5 50 341 276 292 127 379 323)(6 49 342 275 293 126 380 322)(7 48 343 274 294 125 381 321)(8 47 344 273 295 124 382 320)(9 46 345 272 296 123 383 319)(10 45 346 271 297 122 384 318)(11 44 347 270 298 121 385 317)(12 43 348 269 299 120 386 316)(13 42 349 268 300 119 387 315)(14 41 350 267 301 118 388 314)(15 40 351 266 302 117 389 313)(16 39 352 265 303 116 390 338)(17 38 353 264 304 115 365 337)(18 37 354 263 305 114 366 336)(19 36 355 262 306 113 367 335)(20 35 356 261 307 112 368 334)(21 34 357 286 308 111 369 333)(22 33 358 285 309 110 370 332)(23 32 359 284 310 109 371 331)(24 31 360 283 311 108 372 330)(25 30 361 282 312 107 373 329)(26 29 362 281 287 106 374 328)(53 151 202 392 258 227 172 88)(54 150 203 391 259 226 173 87)(55 149 204 416 260 225 174 86)(56 148 205 415 235 224 175 85)(57 147 206 414 236 223 176 84)(58 146 207 413 237 222 177 83)(59 145 208 412 238 221 178 82)(60 144 183 411 239 220 179 81)(61 143 184 410 240 219 180 80)(62 142 185 409 241 218 181 79)(63 141 186 408 242 217 182 104)(64 140 187 407 243 216 157 103)(65 139 188 406 244 215 158 102)(66 138 189 405 245 214 159 101)(67 137 190 404 246 213 160 100)(68 136 191 403 247 212 161 99)(69 135 192 402 248 211 162 98)(70 134 193 401 249 210 163 97)(71 133 194 400 250 209 164 96)(72 132 195 399 251 234 165 95)(73 131 196 398 252 233 166 94)(74 156 197 397 253 232 167 93)(75 155 198 396 254 231 168 92)(76 154 199 395 255 230 169 91)(77 153 200 394 256 229 170 90)(78 152 201 393 257 228 171 89)
(1 253 301 61)(2 254 302 62)(3 255 303 63)(4 256 304 64)(5 257 305 65)(6 258 306 66)(7 259 307 67)(8 260 308 68)(9 235 309 69)(10 236 310 70)(11 237 311 71)(12 238 312 72)(13 239 287 73)(14 240 288 74)(15 241 289 75)(16 242 290 76)(17 243 291 77)(18 244 292 78)(19 245 293 53)(20 246 294 54)(21 247 295 55)(22 248 296 56)(23 249 297 57)(24 250 298 58)(25 251 299 59)(26 252 300 60)(27 409 117 92)(28 410 118 93)(29 411 119 94)(30 412 120 95)(31 413 121 96)(32 414 122 97)(33 415 123 98)(34 416 124 99)(35 391 125 100)(36 392 126 101)(37 393 127 102)(38 394 128 103)(39 395 129 104)(40 396 130 79)(41 397 105 80)(42 398 106 81)(43 399 107 82)(44 400 108 83)(45 401 109 84)(46 402 110 85)(47 403 111 86)(48 404 112 87)(49 405 113 88)(50 406 114 89)(51 407 115 90)(52 408 116 91)(131 328 220 268)(132 329 221 269)(133 330 222 270)(134 331 223 271)(135 332 224 272)(136 333 225 273)(137 334 226 274)(138 335 227 275)(139 336 228 276)(140 337 229 277)(141 338 230 278)(142 313 231 279)(143 314 232 280)(144 315 233 281)(145 316 234 282)(146 317 209 283)(147 318 210 284)(148 319 211 285)(149 320 212 286)(150 321 213 261)(151 322 214 262)(152 323 215 263)(153 324 216 264)(154 325 217 265)(155 326 218 266)(156 327 219 267)(157 340 200 365)(158 341 201 366)(159 342 202 367)(160 343 203 368)(161 344 204 369)(162 345 205 370)(163 346 206 371)(164 347 207 372)(165 348 208 373)(166 349 183 374)(167 350 184 375)(168 351 185 376)(169 352 186 377)(170 353 187 378)(171 354 188 379)(172 355 189 380)(173 356 190 381)(174 357 191 382)(175 358 192 383)(176 359 193 384)(177 360 194 385)(178 361 195 386)(179 362 196 387)(180 363 197 388)(181 364 198 389)(182 339 199 390)
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), (1,28,363,280,288,105,375,327)(2,27,364,279,289,130,376,326)(3,52,339,278,290,129,377,325)(4,51,340,277,291,128,378,324)(5,50,341,276,292,127,379,323)(6,49,342,275,293,126,380,322)(7,48,343,274,294,125,381,321)(8,47,344,273,295,124,382,320)(9,46,345,272,296,123,383,319)(10,45,346,271,297,122,384,318)(11,44,347,270,298,121,385,317)(12,43,348,269,299,120,386,316)(13,42,349,268,300,119,387,315)(14,41,350,267,301,118,388,314)(15,40,351,266,302,117,389,313)(16,39,352,265,303,116,390,338)(17,38,353,264,304,115,365,337)(18,37,354,263,305,114,366,336)(19,36,355,262,306,113,367,335)(20,35,356,261,307,112,368,334)(21,34,357,286,308,111,369,333)(22,33,358,285,309,110,370,332)(23,32,359,284,310,109,371,331)(24,31,360,283,311,108,372,330)(25,30,361,282,312,107,373,329)(26,29,362,281,287,106,374,328)(53,151,202,392,258,227,172,88)(54,150,203,391,259,226,173,87)(55,149,204,416,260,225,174,86)(56,148,205,415,235,224,175,85)(57,147,206,414,236,223,176,84)(58,146,207,413,237,222,177,83)(59,145,208,412,238,221,178,82)(60,144,183,411,239,220,179,81)(61,143,184,410,240,219,180,80)(62,142,185,409,241,218,181,79)(63,141,186,408,242,217,182,104)(64,140,187,407,243,216,157,103)(65,139,188,406,244,215,158,102)(66,138,189,405,245,214,159,101)(67,137,190,404,246,213,160,100)(68,136,191,403,247,212,161,99)(69,135,192,402,248,211,162,98)(70,134,193,401,249,210,163,97)(71,133,194,400,250,209,164,96)(72,132,195,399,251,234,165,95)(73,131,196,398,252,233,166,94)(74,156,197,397,253,232,167,93)(75,155,198,396,254,231,168,92)(76,154,199,395,255,230,169,91)(77,153,200,394,256,229,170,90)(78,152,201,393,257,228,171,89), (1,253,301,61)(2,254,302,62)(3,255,303,63)(4,256,304,64)(5,257,305,65)(6,258,306,66)(7,259,307,67)(8,260,308,68)(9,235,309,69)(10,236,310,70)(11,237,311,71)(12,238,312,72)(13,239,287,73)(14,240,288,74)(15,241,289,75)(16,242,290,76)(17,243,291,77)(18,244,292,78)(19,245,293,53)(20,246,294,54)(21,247,295,55)(22,248,296,56)(23,249,297,57)(24,250,298,58)(25,251,299,59)(26,252,300,60)(27,409,117,92)(28,410,118,93)(29,411,119,94)(30,412,120,95)(31,413,121,96)(32,414,122,97)(33,415,123,98)(34,416,124,99)(35,391,125,100)(36,392,126,101)(37,393,127,102)(38,394,128,103)(39,395,129,104)(40,396,130,79)(41,397,105,80)(42,398,106,81)(43,399,107,82)(44,400,108,83)(45,401,109,84)(46,402,110,85)(47,403,111,86)(48,404,112,87)(49,405,113,88)(50,406,114,89)(51,407,115,90)(52,408,116,91)(131,328,220,268)(132,329,221,269)(133,330,222,270)(134,331,223,271)(135,332,224,272)(136,333,225,273)(137,334,226,274)(138,335,227,275)(139,336,228,276)(140,337,229,277)(141,338,230,278)(142,313,231,279)(143,314,232,280)(144,315,233,281)(145,316,234,282)(146,317,209,283)(147,318,210,284)(148,319,211,285)(149,320,212,286)(150,321,213,261)(151,322,214,262)(152,323,215,263)(153,324,216,264)(154,325,217,265)(155,326,218,266)(156,327,219,267)(157,340,200,365)(158,341,201,366)(159,342,202,367)(160,343,203,368)(161,344,204,369)(162,345,205,370)(163,346,206,371)(164,347,207,372)(165,348,208,373)(166,349,183,374)(167,350,184,375)(168,351,185,376)(169,352,186,377)(170,353,187,378)(171,354,188,379)(172,355,189,380)(173,356,190,381)(174,357,191,382)(175,358,192,383)(176,359,193,384)(177,360,194,385)(178,361,195,386)(179,362,196,387)(180,363,197,388)(181,364,198,389)(182,339,199,390)>;
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), (1,28,363,280,288,105,375,327)(2,27,364,279,289,130,376,326)(3,52,339,278,290,129,377,325)(4,51,340,277,291,128,378,324)(5,50,341,276,292,127,379,323)(6,49,342,275,293,126,380,322)(7,48,343,274,294,125,381,321)(8,47,344,273,295,124,382,320)(9,46,345,272,296,123,383,319)(10,45,346,271,297,122,384,318)(11,44,347,270,298,121,385,317)(12,43,348,269,299,120,386,316)(13,42,349,268,300,119,387,315)(14,41,350,267,301,118,388,314)(15,40,351,266,302,117,389,313)(16,39,352,265,303,116,390,338)(17,38,353,264,304,115,365,337)(18,37,354,263,305,114,366,336)(19,36,355,262,306,113,367,335)(20,35,356,261,307,112,368,334)(21,34,357,286,308,111,369,333)(22,33,358,285,309,110,370,332)(23,32,359,284,310,109,371,331)(24,31,360,283,311,108,372,330)(25,30,361,282,312,107,373,329)(26,29,362,281,287,106,374,328)(53,151,202,392,258,227,172,88)(54,150,203,391,259,226,173,87)(55,149,204,416,260,225,174,86)(56,148,205,415,235,224,175,85)(57,147,206,414,236,223,176,84)(58,146,207,413,237,222,177,83)(59,145,208,412,238,221,178,82)(60,144,183,411,239,220,179,81)(61,143,184,410,240,219,180,80)(62,142,185,409,241,218,181,79)(63,141,186,408,242,217,182,104)(64,140,187,407,243,216,157,103)(65,139,188,406,244,215,158,102)(66,138,189,405,245,214,159,101)(67,137,190,404,246,213,160,100)(68,136,191,403,247,212,161,99)(69,135,192,402,248,211,162,98)(70,134,193,401,249,210,163,97)(71,133,194,400,250,209,164,96)(72,132,195,399,251,234,165,95)(73,131,196,398,252,233,166,94)(74,156,197,397,253,232,167,93)(75,155,198,396,254,231,168,92)(76,154,199,395,255,230,169,91)(77,153,200,394,256,229,170,90)(78,152,201,393,257,228,171,89), (1,253,301,61)(2,254,302,62)(3,255,303,63)(4,256,304,64)(5,257,305,65)(6,258,306,66)(7,259,307,67)(8,260,308,68)(9,235,309,69)(10,236,310,70)(11,237,311,71)(12,238,312,72)(13,239,287,73)(14,240,288,74)(15,241,289,75)(16,242,290,76)(17,243,291,77)(18,244,292,78)(19,245,293,53)(20,246,294,54)(21,247,295,55)(22,248,296,56)(23,249,297,57)(24,250,298,58)(25,251,299,59)(26,252,300,60)(27,409,117,92)(28,410,118,93)(29,411,119,94)(30,412,120,95)(31,413,121,96)(32,414,122,97)(33,415,123,98)(34,416,124,99)(35,391,125,100)(36,392,126,101)(37,393,127,102)(38,394,128,103)(39,395,129,104)(40,396,130,79)(41,397,105,80)(42,398,106,81)(43,399,107,82)(44,400,108,83)(45,401,109,84)(46,402,110,85)(47,403,111,86)(48,404,112,87)(49,405,113,88)(50,406,114,89)(51,407,115,90)(52,408,116,91)(131,328,220,268)(132,329,221,269)(133,330,222,270)(134,331,223,271)(135,332,224,272)(136,333,225,273)(137,334,226,274)(138,335,227,275)(139,336,228,276)(140,337,229,277)(141,338,230,278)(142,313,231,279)(143,314,232,280)(144,315,233,281)(145,316,234,282)(146,317,209,283)(147,318,210,284)(148,319,211,285)(149,320,212,286)(150,321,213,261)(151,322,214,262)(152,323,215,263)(153,324,216,264)(154,325,217,265)(155,326,218,266)(156,327,219,267)(157,340,200,365)(158,341,201,366)(159,342,202,367)(160,343,203,368)(161,344,204,369)(162,345,205,370)(163,346,206,371)(164,347,207,372)(165,348,208,373)(166,349,183,374)(167,350,184,375)(168,351,185,376)(169,352,186,377)(170,353,187,378)(171,354,188,379)(172,355,189,380)(173,356,190,381)(174,357,191,382)(175,358,192,383)(176,359,193,384)(177,360,194,385)(178,361,195,386)(179,362,196,387)(180,363,197,388)(181,364,198,389)(182,339,199,390) );
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)], [(1,28,363,280,288,105,375,327),(2,27,364,279,289,130,376,326),(3,52,339,278,290,129,377,325),(4,51,340,277,291,128,378,324),(5,50,341,276,292,127,379,323),(6,49,342,275,293,126,380,322),(7,48,343,274,294,125,381,321),(8,47,344,273,295,124,382,320),(9,46,345,272,296,123,383,319),(10,45,346,271,297,122,384,318),(11,44,347,270,298,121,385,317),(12,43,348,269,299,120,386,316),(13,42,349,268,300,119,387,315),(14,41,350,267,301,118,388,314),(15,40,351,266,302,117,389,313),(16,39,352,265,303,116,390,338),(17,38,353,264,304,115,365,337),(18,37,354,263,305,114,366,336),(19,36,355,262,306,113,367,335),(20,35,356,261,307,112,368,334),(21,34,357,286,308,111,369,333),(22,33,358,285,309,110,370,332),(23,32,359,284,310,109,371,331),(24,31,360,283,311,108,372,330),(25,30,361,282,312,107,373,329),(26,29,362,281,287,106,374,328),(53,151,202,392,258,227,172,88),(54,150,203,391,259,226,173,87),(55,149,204,416,260,225,174,86),(56,148,205,415,235,224,175,85),(57,147,206,414,236,223,176,84),(58,146,207,413,237,222,177,83),(59,145,208,412,238,221,178,82),(60,144,183,411,239,220,179,81),(61,143,184,410,240,219,180,80),(62,142,185,409,241,218,181,79),(63,141,186,408,242,217,182,104),(64,140,187,407,243,216,157,103),(65,139,188,406,244,215,158,102),(66,138,189,405,245,214,159,101),(67,137,190,404,246,213,160,100),(68,136,191,403,247,212,161,99),(69,135,192,402,248,211,162,98),(70,134,193,401,249,210,163,97),(71,133,194,400,250,209,164,96),(72,132,195,399,251,234,165,95),(73,131,196,398,252,233,166,94),(74,156,197,397,253,232,167,93),(75,155,198,396,254,231,168,92),(76,154,199,395,255,230,169,91),(77,153,200,394,256,229,170,90),(78,152,201,393,257,228,171,89)], [(1,253,301,61),(2,254,302,62),(3,255,303,63),(4,256,304,64),(5,257,305,65),(6,258,306,66),(7,259,307,67),(8,260,308,68),(9,235,309,69),(10,236,310,70),(11,237,311,71),(12,238,312,72),(13,239,287,73),(14,240,288,74),(15,241,289,75),(16,242,290,76),(17,243,291,77),(18,244,292,78),(19,245,293,53),(20,246,294,54),(21,247,295,55),(22,248,296,56),(23,249,297,57),(24,250,298,58),(25,251,299,59),(26,252,300,60),(27,409,117,92),(28,410,118,93),(29,411,119,94),(30,412,120,95),(31,413,121,96),(32,414,122,97),(33,415,123,98),(34,416,124,99),(35,391,125,100),(36,392,126,101),(37,393,127,102),(38,394,128,103),(39,395,129,104),(40,396,130,79),(41,397,105,80),(42,398,106,81),(43,399,107,82),(44,400,108,83),(45,401,109,84),(46,402,110,85),(47,403,111,86),(48,404,112,87),(49,405,113,88),(50,406,114,89),(51,407,115,90),(52,408,116,91),(131,328,220,268),(132,329,221,269),(133,330,222,270),(134,331,223,271),(135,332,224,272),(136,333,225,273),(137,334,226,274),(138,335,227,275),(139,336,228,276),(140,337,229,277),(141,338,230,278),(142,313,231,279),(143,314,232,280),(144,315,233,281),(145,316,234,282),(146,317,209,283),(147,318,210,284),(148,319,211,285),(149,320,212,286),(150,321,213,261),(151,322,214,262),(152,323,215,263),(153,324,216,264),(154,325,217,265),(155,326,218,266),(156,327,219,267),(157,340,200,365),(158,341,201,366),(159,342,202,367),(160,343,203,368),(161,344,204,369),(162,345,205,370),(163,346,206,371),(164,347,207,372),(165,348,208,373),(166,349,183,374),(167,350,184,375),(168,351,185,376),(169,352,186,377),(170,353,187,378),(171,354,188,379),(172,355,189,380),(173,356,190,381),(174,357,191,382),(175,358,192,383),(176,359,193,384),(177,360,194,385),(178,361,195,386),(179,362,196,387),(180,363,197,388),(181,364,198,389),(182,339,199,390)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26R | 52A | ··· | 52AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 52 | 52 | 26 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | SD16 | Q16 | D13 | D26 | C4×D13 | D52 | C13⋊D4 | D4.D13 | C13⋊Q16 |
| kernel | C26.Q16 | C2×C13⋊2C8 | C13×C4⋊C4 | C2×Dic26 | Dic26 | C52 | C2×C26 | C26 | C26 | C4⋊C4 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of C26.Q16 ►in GL4(𝔽313) generated by
| 307 | 307 | 0 | 0 |
| 6 | 58 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 290 | 9 | 0 | 0 |
| 150 | 23 | 0 | 0 |
| 0 | 0 | 0 | 61 |
| 0 | 0 | 118 | 120 |
| 159 | 245 | 0 | 0 |
| 68 | 154 | 0 | 0 |
| 0 | 0 | 197 | 36 |
| 0 | 0 | 87 | 116 |
G:=sub<GL(4,GF(313))| [307,6,0,0,307,58,0,0,0,0,1,0,0,0,0,1],[290,150,0,0,9,23,0,0,0,0,0,118,0,0,61,120],[159,68,0,0,245,154,0,0,0,0,197,87,0,0,36,116] >;
C26.Q16 in GAP, Magma, Sage, TeX
C_{26}.Q_{16} % in TeX
G:=Group("C26.Q16"); // GroupNames label
G:=SmallGroup(416,17);
// by ID
G=gap.SmallGroup(416,17);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,121,31,579,297,69,13829]);
// Polycyclic
G:=Group<a,b,c|a^26=b^8=1,c^2=a^13*b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^13*b^-1>;
// generators/relations
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