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G = C13×D13order 338 = 2·132

Direct product of C13 and D13

direct product, metacyclic, supersoluble, monomial, A-group

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

Series: Derived Chief Lower central Upper central

C1C13 — C13×D13
C1C13C132 — C13×D13
C13 — C13×D13
C1C13

Generators and relations for C13×D13
 G = < a,b,c | a13=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

13C2
2C13
2C13
2C13
2C13
2C13
2C13
13C26

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 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)

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), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)>;

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), (1,14)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26) );

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)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26)])

G:=TransitiveGroup(26,11);

104 conjugacy classes

class 1  2 13A···13L13M···13CL26A···26L
order1213···1313···1326···26
size1131···12···213···13

104 irreducible representations

dim111122
type+++
imageC1C2C13C26D13C13×D13
kernelC13×D13C132D13C13C13C1
# reps111212672

Matrix representation of C13×D13 in GL2(𝔽53) generated by

360
036
,
460
015
,
015
460
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

Subgroup lattice of C13×D13 in TeX

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