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## G = C13×D13order 338 = 2·132

### Direct product of C13 and D13

Aliases: C13×D13, C13C2, AΣL1(𝔽169), C13⋊C26, C1321C2, SmallGroup(338,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C13×D13
 Chief series C1 — C13 — C132 — C13×D13
 Lower central C13 — C13×D13
 Upper central C1 — C13

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

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