direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C13×D13, C13≀C2, AΣL1(𝔽169), C13⋊C26, C132⋊1C2, SmallGroup(338,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C13×D13 |
Generators and relations for C13×D13
G = < a,b,c | a13=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Permutation representations of C13×D13
►On 26 points - transitive group 26T11
Generators in S26
(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)
(1 7 13 6 12 5 11 4 10 3 9 2 8)(14 21 15 22 16 23 17 24 18 25 19 26 20)
(1 24)(2 25)(3 26)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)
G:=sub<Sym(26)| (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), (1,7,13,6,12,5,11,4,10,3,9,2,8)(14,21,15,22,16,23,17,24,18,25,19,26,20), (1,24)(2,25)(3,26)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)>;
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), (1,7,13,6,12,5,11,4,10,3,9,2,8)(14,21,15,22,16,23,17,24,18,25,19,26,20), (1,24)(2,25)(3,26)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23) );
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)], [(1,7,13,6,12,5,11,4,10,3,9,2,8),(14,21,15,22,16,23,17,24,18,25,19,26,20)], [(1,24),(2,25),(3,26),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(13,23)]])
G:=TransitiveGroup(26,11);
104 conjugacy classes
| class | 1 | 2 | 13A | ··· | 13L | 13M | ··· | 13CL | 26A | ··· | 26L |
| order | 1 | 2 | 13 | ··· | 13 | 13 | ··· | 13 | 26 | ··· | 26 |
| size | 1 | 13 | 1 | ··· | 1 | 2 | ··· | 2 | 13 | ··· | 13 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C13 | C26 | D13 | C13×D13 |
| kernel | C13×D13 | C132 | D13 | C13 | C13 | C1 |
| # reps | 1 | 1 | 12 | 12 | 6 | 72 |
Matrix representation of C13×D13 ►in GL2(𝔽53) generated by
| 36 | 0 |
| 0 | 36 |
| 46 | 0 |
| 0 | 15 |
| 0 | 15 |
| 46 | 0 |
G:=sub<GL(2,GF(53))| [36,0,0,36],[46,0,0,15],[0,46,15,0] >;
C13×D13 in GAP, Magma, Sage, TeX
C_{13}\times D_{13} % in TeX
G:=Group("C13xD13"); // GroupNames label
G:=SmallGroup(338,3);
// by ID
G=gap.SmallGroup(338,3);
# by ID
G:=PCGroup([3,-2,-13,-13,2810]);
// Polycyclic
G:=Group<a,b,c|a^13=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export