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G = C2×C13⋊C6  order 156 = 22·3·13

Direct product of C2 and C13⋊C6

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

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 >

C1
C2
13C2
13C2
13C3
C13
13C22
13C6
13C6
13C6
D13
D13
C26
C13⋊C3
13C2×C6
D26
C13⋊C6
C13⋊C6
C2×C13⋊C3
C2×C13⋊C6

Character table of C2×C13⋊C6

 class 12A2B2C3A3B6A6B6C6D6E6F13A13B26A26B
 size 11131313131313131313136666
ρ11111111111111111    trivial
ρ211-1-111-1-1-11-111111    linear of order 2
ρ31-1-11111-1-1-11-111-1-1    linear of order 2
ρ41-11-111-111-1-1-111-1-1    linear of order 2
ρ511-1-1ζ32ζ3ζ6ζ65ζ6ζ32ζ65ζ31111    linear of order 6
ρ61111ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ31111    linear of order 3
ρ71-11-1ζ3ζ32ζ65ζ32ζ3ζ65ζ6ζ611-1-1    linear of order 6
ρ81111ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ321111    linear of order 3
ρ91-1-11ζ32ζ3ζ32ζ65ζ6ζ6ζ3ζ6511-1-1    linear of order 6
ρ101-11-1ζ32ζ3ζ6ζ3ζ32ζ6ζ65ζ6511-1-1    linear of order 6
ρ1111-1-1ζ3ζ32ζ65ζ6ζ65ζ3ζ6ζ321111    linear of order 6
ρ121-1-11ζ3ζ32ζ3ζ6ζ65ζ65ζ32ζ611-1-1    linear of order 6
ρ136-60000000000-1+√13/2-1-√13/21+√13/21-√13/2    orthogonal faithful
ρ146-60000000000-1-√13/2-1+√13/21-√13/21+√13/2    orthogonal faithful
ρ15660000000000-1-√13/2-1+√13/2-1+√13/2-1-√13/2    orthogonal lifted from C13⋊C6
ρ16660000000000-1+√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)

200000
020000
002000
000200
000020
000002
,
001000
000120
000100
010000
000202
200100
,
102020
000110
021020
001011
000010
002020

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

Subgroup lattice of C2×C13⋊C6 in TeX
Character table of C2×C13⋊C6 in TeX

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