Copied to
clipboard

## G = C2×C13⋊C6order 156 = 22·3·13

### Direct product of C2 and C13⋊C6

Aliases: C2×C13⋊C6, C26⋊C6, D26⋊C3, D13⋊C6, C13⋊(C2×C6), C13⋊C3⋊C22, (C2×C13⋊C3)⋊C2, SmallGroup(156,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C2×C13⋊C6
 Chief series C1 — C13 — C13⋊C3 — C13⋊C6 — C2×C13⋊C6
 Lower central C13 — C2×C13⋊C6
 Upper central C1 — C2

Generators and relations for C2×C13⋊C6
G = < a,b,c | a2=b13=c6=1, ab=ba, ac=ca, cbc-1=b10 >

Character table of C2×C13⋊C6

 class 1 2A 2B 2C 3A 3B 6A 6B 6C 6D 6E 6F 13A 13B 26A 26B size 1 1 13 13 13 13 13 13 13 13 13 13 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 linear of order 2 ρ3 1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ4 1 -1 1 -1 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ5 1 1 -1 -1 ζ32 ζ3 ζ6 ζ65 ζ6 ζ32 ζ65 ζ3 1 1 1 1 linear of order 6 ρ6 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 1 1 linear of order 3 ρ7 1 -1 1 -1 ζ3 ζ32 ζ65 ζ32 ζ3 ζ65 ζ6 ζ6 1 1 -1 -1 linear of order 6 ρ8 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 1 1 linear of order 3 ρ9 1 -1 -1 1 ζ32 ζ3 ζ32 ζ65 ζ6 ζ6 ζ3 ζ65 1 1 -1 -1 linear of order 6 ρ10 1 -1 1 -1 ζ32 ζ3 ζ6 ζ3 ζ32 ζ6 ζ65 ζ65 1 1 -1 -1 linear of order 6 ρ11 1 1 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ65 ζ3 ζ6 ζ32 1 1 1 1 linear of order 6 ρ12 1 -1 -1 1 ζ3 ζ32 ζ3 ζ6 ζ65 ζ65 ζ32 ζ6 1 1 -1 -1 linear of order 6 ρ13 6 -6 0 0 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 1+√13/2 1-√13/2 orthogonal faithful ρ14 6 -6 0 0 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 1-√13/2 1+√13/2 orthogonal faithful ρ15 6 6 0 0 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 orthogonal lifted from C13⋊C6 ρ16 6 6 0 0 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 orthogonal lifted from C13⋊C6

Permutation representations of C2×C13⋊C6
On 26 points - transitive group 26T9
Generators in S26
(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)
(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 14)(2 18 4 26 10 24)(3 22 7 25 6 21)(5 17 13 23 11 15)(8 16 9 20 12 19)

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

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

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

G:=TransitiveGroup(26,9);

C2×C13⋊C6 is a maximal subgroup of   D52⋊C3  D26⋊C6
C2×C13⋊C6 is a maximal quotient of   Dic26⋊C3  D52⋊C3  D26⋊C6

Matrix representation of C2×C13⋊C6 in GL6(𝔽3)

 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2
,
 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 2 0 2 2 0 0 1 0 0
,
 1 0 2 0 2 0 0 0 0 1 1 0 0 2 1 0 2 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 2 0 2 0

G:=sub<GL(6,GF(3))| [2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[0,0,0,0,0,2,0,0,0,1,0,0,1,0,0,0,0,0,0,1,1,0,2,1,0,2,0,0,0,0,0,0,0,0,2,0],[1,0,0,0,0,0,0,0,2,0,0,0,2,0,1,1,0,2,0,1,0,0,0,0,2,1,2,1,1,2,0,0,0,1,0,0] >;

C2×C13⋊C6 in GAP, Magma, Sage, TeX

C_2\times C_{13}\rtimes C_6
% in TeX

G:=Group("C2xC13:C6");
// GroupNames label

G:=SmallGroup(156,8);
// by ID

G=gap.SmallGroup(156,8);
# by ID

G:=PCGroup([4,-2,-2,-3,-13,2307,155]);
// Polycyclic

G:=Group<a,b,c|a^2=b^13=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations

Export

׿
×
𝔽