direct product, abelian, monomial, 13-elementary
Aliases: C13×C26, SmallGroup(338,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C13×C26 |
| C1 — C13×C26 |
| C1 — C13×C26 |
Generators and relations for C13×C26
G = < a,b | a13=b26=1, ab=ba >
Smallest permutation representation of C13×C26
►Regular action on 338 points
Generators in S338
(1 191 308 320 173 272 47 91 74 123 237 133 224)(2 192 309 321 174 273 48 92 75 124 238 134 225)(3 193 310 322 175 274 49 93 76 125 239 135 226)(4 194 311 323 176 275 50 94 77 126 240 136 227)(5 195 312 324 177 276 51 95 78 127 241 137 228)(6 196 287 325 178 277 52 96 53 128 242 138 229)(7 197 288 326 179 278 27 97 54 129 243 139 230)(8 198 289 327 180 279 28 98 55 130 244 140 231)(9 199 290 328 181 280 29 99 56 105 245 141 232)(10 200 291 329 182 281 30 100 57 106 246 142 233)(11 201 292 330 157 282 31 101 58 107 247 143 234)(12 202 293 331 158 283 32 102 59 108 248 144 209)(13 203 294 332 159 284 33 103 60 109 249 145 210)(14 204 295 333 160 285 34 104 61 110 250 146 211)(15 205 296 334 161 286 35 79 62 111 251 147 212)(16 206 297 335 162 261 36 80 63 112 252 148 213)(17 207 298 336 163 262 37 81 64 113 253 149 214)(18 208 299 337 164 263 38 82 65 114 254 150 215)(19 183 300 338 165 264 39 83 66 115 255 151 216)(20 184 301 313 166 265 40 84 67 116 256 152 217)(21 185 302 314 167 266 41 85 68 117 257 153 218)(22 186 303 315 168 267 42 86 69 118 258 154 219)(23 187 304 316 169 268 43 87 70 119 259 155 220)(24 188 305 317 170 269 44 88 71 120 260 156 221)(25 189 306 318 171 270 45 89 72 121 235 131 222)(26 190 307 319 172 271 46 90 73 122 236 132 223)
(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)
G:=sub<Sym(338)| (1,191,308,320,173,272,47,91,74,123,237,133,224)(2,192,309,321,174,273,48,92,75,124,238,134,225)(3,193,310,322,175,274,49,93,76,125,239,135,226)(4,194,311,323,176,275,50,94,77,126,240,136,227)(5,195,312,324,177,276,51,95,78,127,241,137,228)(6,196,287,325,178,277,52,96,53,128,242,138,229)(7,197,288,326,179,278,27,97,54,129,243,139,230)(8,198,289,327,180,279,28,98,55,130,244,140,231)(9,199,290,328,181,280,29,99,56,105,245,141,232)(10,200,291,329,182,281,30,100,57,106,246,142,233)(11,201,292,330,157,282,31,101,58,107,247,143,234)(12,202,293,331,158,283,32,102,59,108,248,144,209)(13,203,294,332,159,284,33,103,60,109,249,145,210)(14,204,295,333,160,285,34,104,61,110,250,146,211)(15,205,296,334,161,286,35,79,62,111,251,147,212)(16,206,297,335,162,261,36,80,63,112,252,148,213)(17,207,298,336,163,262,37,81,64,113,253,149,214)(18,208,299,337,164,263,38,82,65,114,254,150,215)(19,183,300,338,165,264,39,83,66,115,255,151,216)(20,184,301,313,166,265,40,84,67,116,256,152,217)(21,185,302,314,167,266,41,85,68,117,257,153,218)(22,186,303,315,168,267,42,86,69,118,258,154,219)(23,187,304,316,169,268,43,87,70,119,259,155,220)(24,188,305,317,170,269,44,88,71,120,260,156,221)(25,189,306,318,171,270,45,89,72,121,235,131,222)(26,190,307,319,172,271,46,90,73,122,236,132,223), (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)>;
G:=Group( (1,191,308,320,173,272,47,91,74,123,237,133,224)(2,192,309,321,174,273,48,92,75,124,238,134,225)(3,193,310,322,175,274,49,93,76,125,239,135,226)(4,194,311,323,176,275,50,94,77,126,240,136,227)(5,195,312,324,177,276,51,95,78,127,241,137,228)(6,196,287,325,178,277,52,96,53,128,242,138,229)(7,197,288,326,179,278,27,97,54,129,243,139,230)(8,198,289,327,180,279,28,98,55,130,244,140,231)(9,199,290,328,181,280,29,99,56,105,245,141,232)(10,200,291,329,182,281,30,100,57,106,246,142,233)(11,201,292,330,157,282,31,101,58,107,247,143,234)(12,202,293,331,158,283,32,102,59,108,248,144,209)(13,203,294,332,159,284,33,103,60,109,249,145,210)(14,204,295,333,160,285,34,104,61,110,250,146,211)(15,205,296,334,161,286,35,79,62,111,251,147,212)(16,206,297,335,162,261,36,80,63,112,252,148,213)(17,207,298,336,163,262,37,81,64,113,253,149,214)(18,208,299,337,164,263,38,82,65,114,254,150,215)(19,183,300,338,165,264,39,83,66,115,255,151,216)(20,184,301,313,166,265,40,84,67,116,256,152,217)(21,185,302,314,167,266,41,85,68,117,257,153,218)(22,186,303,315,168,267,42,86,69,118,258,154,219)(23,187,304,316,169,268,43,87,70,119,259,155,220)(24,188,305,317,170,269,44,88,71,120,260,156,221)(25,189,306,318,171,270,45,89,72,121,235,131,222)(26,190,307,319,172,271,46,90,73,122,236,132,223), (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) );
G=PermutationGroup([[(1,191,308,320,173,272,47,91,74,123,237,133,224),(2,192,309,321,174,273,48,92,75,124,238,134,225),(3,193,310,322,175,274,49,93,76,125,239,135,226),(4,194,311,323,176,275,50,94,77,126,240,136,227),(5,195,312,324,177,276,51,95,78,127,241,137,228),(6,196,287,325,178,277,52,96,53,128,242,138,229),(7,197,288,326,179,278,27,97,54,129,243,139,230),(8,198,289,327,180,279,28,98,55,130,244,140,231),(9,199,290,328,181,280,29,99,56,105,245,141,232),(10,200,291,329,182,281,30,100,57,106,246,142,233),(11,201,292,330,157,282,31,101,58,107,247,143,234),(12,202,293,331,158,283,32,102,59,108,248,144,209),(13,203,294,332,159,284,33,103,60,109,249,145,210),(14,204,295,333,160,285,34,104,61,110,250,146,211),(15,205,296,334,161,286,35,79,62,111,251,147,212),(16,206,297,335,162,261,36,80,63,112,252,148,213),(17,207,298,336,163,262,37,81,64,113,253,149,214),(18,208,299,337,164,263,38,82,65,114,254,150,215),(19,183,300,338,165,264,39,83,66,115,255,151,216),(20,184,301,313,166,265,40,84,67,116,256,152,217),(21,185,302,314,167,266,41,85,68,117,257,153,218),(22,186,303,315,168,267,42,86,69,118,258,154,219),(23,187,304,316,169,268,43,87,70,119,259,155,220),(24,188,305,317,170,269,44,88,71,120,260,156,221),(25,189,306,318,171,270,45,89,72,121,235,131,222),(26,190,307,319,172,271,46,90,73,122,236,132,223)], [(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)]])
338 conjugacy classes
| class | 1 | 2 | 13A | ··· | 13FL | 26A | ··· | 26FL |
| order | 1 | 2 | 13 | ··· | 13 | 26 | ··· | 26 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
338 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C13 | C26 |
| kernel | C13×C26 | C132 | C26 | C13 |
| # reps | 1 | 1 | 168 | 168 |
Matrix representation of C13×C26 ►in GL2(𝔽53) generated by
| 47 | 0 |
| 0 | 28 |
| 1 | 0 |
| 0 | 7 |
G:=sub<GL(2,GF(53))| [47,0,0,28],[1,0,0,7] >;
C13×C26 in GAP, Magma, Sage, TeX
C_{13}\times C_{26} % in TeX
G:=Group("C13xC26"); // GroupNames label
G:=SmallGroup(338,5);
// by ID
G=gap.SmallGroup(338,5);
# by ID
G:=PCGroup([3,-2,-13,-13]);
// Polycyclic
G:=Group<a,b|a^13=b^26=1,a*b=b*a>;
// generators/relations
Export