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G = C13×C26order 338 = 2·132

Abelian group of type [13,26]

direct product, abelian, monomial, 13-elementary

Aliases: C13×C26, SmallGroup(338,5)

Series: Derived Chief Lower central Upper central

C1 — C13×C26
C1C13C132 — 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
(1 155 300 317 192 103 281 116 160 27 244 211 77)(2 156 301 318 193 104 282 117 161 28 245 212 78)(3 131 302 319 194 79 283 118 162 29 246 213 53)(4 132 303 320 195 80 284 119 163 30 247 214 54)(5 133 304 321 196 81 285 120 164 31 248 215 55)(6 134 305 322 197 82 286 121 165 32 249 216 56)(7 135 306 323 198 83 261 122 166 33 250 217 57)(8 136 307 324 199 84 262 123 167 34 251 218 58)(9 137 308 325 200 85 263 124 168 35 252 219 59)(10 138 309 326 201 86 264 125 169 36 253 220 60)(11 139 310 327 202 87 265 126 170 37 254 221 61)(12 140 311 328 203 88 266 127 171 38 255 222 62)(13 141 312 329 204 89 267 128 172 39 256 223 63)(14 142 287 330 205 90 268 129 173 40 257 224 64)(15 143 288 331 206 91 269 130 174 41 258 225 65)(16 144 289 332 207 92 270 105 175 42 259 226 66)(17 145 290 333 208 93 271 106 176 43 260 227 67)(18 146 291 334 183 94 272 107 177 44 235 228 68)(19 147 292 335 184 95 273 108 178 45 236 229 69)(20 148 293 336 185 96 274 109 179 46 237 230 70)(21 149 294 337 186 97 275 110 180 47 238 231 71)(22 150 295 338 187 98 276 111 181 48 239 232 72)(23 151 296 313 188 99 277 112 182 49 240 233 73)(24 152 297 314 189 100 278 113 157 50 241 234 74)(25 153 298 315 190 101 279 114 158 51 242 209 75)(26 154 299 316 191 102 280 115 159 52 243 210 76)
(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)| (1,155,300,317,192,103,281,116,160,27,244,211,77)(2,156,301,318,193,104,282,117,161,28,245,212,78)(3,131,302,319,194,79,283,118,162,29,246,213,53)(4,132,303,320,195,80,284,119,163,30,247,214,54)(5,133,304,321,196,81,285,120,164,31,248,215,55)(6,134,305,322,197,82,286,121,165,32,249,216,56)(7,135,306,323,198,83,261,122,166,33,250,217,57)(8,136,307,324,199,84,262,123,167,34,251,218,58)(9,137,308,325,200,85,263,124,168,35,252,219,59)(10,138,309,326,201,86,264,125,169,36,253,220,60)(11,139,310,327,202,87,265,126,170,37,254,221,61)(12,140,311,328,203,88,266,127,171,38,255,222,62)(13,141,312,329,204,89,267,128,172,39,256,223,63)(14,142,287,330,205,90,268,129,173,40,257,224,64)(15,143,288,331,206,91,269,130,174,41,258,225,65)(16,144,289,332,207,92,270,105,175,42,259,226,66)(17,145,290,333,208,93,271,106,176,43,260,227,67)(18,146,291,334,183,94,272,107,177,44,235,228,68)(19,147,292,335,184,95,273,108,178,45,236,229,69)(20,148,293,336,185,96,274,109,179,46,237,230,70)(21,149,294,337,186,97,275,110,180,47,238,231,71)(22,150,295,338,187,98,276,111,181,48,239,232,72)(23,151,296,313,188,99,277,112,182,49,240,233,73)(24,152,297,314,189,100,278,113,157,50,241,234,74)(25,153,298,315,190,101,279,114,158,51,242,209,75)(26,154,299,316,191,102,280,115,159,52,243,210,76), (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( (1,155,300,317,192,103,281,116,160,27,244,211,77)(2,156,301,318,193,104,282,117,161,28,245,212,78)(3,131,302,319,194,79,283,118,162,29,246,213,53)(4,132,303,320,195,80,284,119,163,30,247,214,54)(5,133,304,321,196,81,285,120,164,31,248,215,55)(6,134,305,322,197,82,286,121,165,32,249,216,56)(7,135,306,323,198,83,261,122,166,33,250,217,57)(8,136,307,324,199,84,262,123,167,34,251,218,58)(9,137,308,325,200,85,263,124,168,35,252,219,59)(10,138,309,326,201,86,264,125,169,36,253,220,60)(11,139,310,327,202,87,265,126,170,37,254,221,61)(12,140,311,328,203,88,266,127,171,38,255,222,62)(13,141,312,329,204,89,267,128,172,39,256,223,63)(14,142,287,330,205,90,268,129,173,40,257,224,64)(15,143,288,331,206,91,269,130,174,41,258,225,65)(16,144,289,332,207,92,270,105,175,42,259,226,66)(17,145,290,333,208,93,271,106,176,43,260,227,67)(18,146,291,334,183,94,272,107,177,44,235,228,68)(19,147,292,335,184,95,273,108,178,45,236,229,69)(20,148,293,336,185,96,274,109,179,46,237,230,70)(21,149,294,337,186,97,275,110,180,47,238,231,71)(22,150,295,338,187,98,276,111,181,48,239,232,72)(23,151,296,313,188,99,277,112,182,49,240,233,73)(24,152,297,314,189,100,278,113,157,50,241,234,74)(25,153,298,315,190,101,279,114,158,51,242,209,75)(26,154,299,316,191,102,280,115,159,52,243,210,76), (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([(1,155,300,317,192,103,281,116,160,27,244,211,77),(2,156,301,318,193,104,282,117,161,28,245,212,78),(3,131,302,319,194,79,283,118,162,29,246,213,53),(4,132,303,320,195,80,284,119,163,30,247,214,54),(5,133,304,321,196,81,285,120,164,31,248,215,55),(6,134,305,322,197,82,286,121,165,32,249,216,56),(7,135,306,323,198,83,261,122,166,33,250,217,57),(8,136,307,324,199,84,262,123,167,34,251,218,58),(9,137,308,325,200,85,263,124,168,35,252,219,59),(10,138,309,326,201,86,264,125,169,36,253,220,60),(11,139,310,327,202,87,265,126,170,37,254,221,61),(12,140,311,328,203,88,266,127,171,38,255,222,62),(13,141,312,329,204,89,267,128,172,39,256,223,63),(14,142,287,330,205,90,268,129,173,40,257,224,64),(15,143,288,331,206,91,269,130,174,41,258,225,65),(16,144,289,332,207,92,270,105,175,42,259,226,66),(17,145,290,333,208,93,271,106,176,43,260,227,67),(18,146,291,334,183,94,272,107,177,44,235,228,68),(19,147,292,335,184,95,273,108,178,45,236,229,69),(20,148,293,336,185,96,274,109,179,46,237,230,70),(21,149,294,337,186,97,275,110,180,47,238,231,71),(22,150,295,338,187,98,276,111,181,48,239,232,72),(23,151,296,313,188,99,277,112,182,49,240,233,73),(24,152,297,314,189,100,278,113,157,50,241,234,74),(25,153,298,315,190,101,279,114,158,51,242,209,75),(26,154,299,316,191,102,280,115,159,52,243,210,76)], [(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···13FL26A···26FL
order1213···1326···26
size111···11···1

338 irreducible representations

dim1111
type++
imageC1C2C13C26
kernelC13×C26C132C26C13
# reps11168168

Matrix representation of C13×C26 in GL2(𝔽53) generated by

470
028
,
10
07
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

Subgroup lattice of C13×C26 in TeX

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