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## G = C2×F13order 312 = 23·3·13

### Direct product of C2 and F13

Aliases: C2×F13, C26⋊C12, D13⋊C12, D26.C6, C13⋊C4⋊C6, C13⋊C6⋊C4, C13⋊(C2×C12), D13.(C2×C6), C13⋊C6.C22, (C2×C13⋊C4)⋊C3, C13⋊C3⋊(C2×C4), (C2×C13⋊C3)⋊C4, (C2×C13⋊C6).C2, Aut(D26), Hol(C26), SmallGroup(312,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C2×F13
 Chief series C1 — C13 — D13 — C13⋊C6 — F13 — C2×F13
 Lower central C13 — C2×F13
 Upper central C1 — C2

Generators and relations for C2×F13
G = < a,b,c | a2=b13=c12=1, ab=ba, ac=ca, cbc-1=b6 >

Character table of C2×F13

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H 13 26 size 1 1 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 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 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 -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 -1 -1 -1 -1 1 1 1 1 1 -1 linear of order 2 ρ5 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 linear of order 3 ρ6 1 -1 -1 1 ζ3 ζ32 -1 1 1 -1 ζ6 ζ3 ζ32 ζ65 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 ζ65 ζ65 ζ6 ζ6 1 -1 linear of order 6 ρ7 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ65 ζ65 ζ6 ζ6 1 1 linear of order 6 ρ8 1 -1 -1 1 ζ3 ζ32 1 -1 -1 1 ζ6 ζ3 ζ32 ζ65 ζ65 ζ6 ζ65 ζ65 ζ6 ζ6 ζ3 ζ3 ζ32 ζ32 1 -1 linear of order 6 ρ9 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 linear of order 3 ρ10 1 -1 -1 1 ζ32 ζ3 -1 1 1 -1 ζ65 ζ32 ζ3 ζ6 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 ζ6 ζ6 ζ65 ζ65 1 -1 linear of order 6 ρ11 1 1 1 1 ζ32 ζ3 -1 -1 -1 -1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ6 ζ6 ζ65 ζ65 1 1 linear of order 6 ρ12 1 -1 -1 1 ζ32 ζ3 1 -1 -1 1 ζ65 ζ32 ζ3 ζ6 ζ6 ζ65 ζ6 ζ6 ζ65 ζ65 ζ32 ζ32 ζ3 ζ3 1 -1 linear of order 6 ρ13 1 1 -1 -1 1 1 i -i i -i -1 -1 -1 1 -1 1 -i i -i i -i i -i i 1 1 linear of order 4 ρ14 1 -1 1 -1 1 1 -i -i i i 1 -1 -1 -1 1 -1 -i i -i i i -i i -i 1 -1 linear of order 4 ρ15 1 -1 1 -1 1 1 i i -i -i 1 -1 -1 -1 1 -1 i -i i -i -i i -i i 1 -1 linear of order 4 ρ16 1 1 -1 -1 1 1 -i i -i i -1 -1 -1 1 -1 1 i -i i -i i -i i -i 1 1 linear of order 4 ρ17 1 1 -1 -1 ζ3 ζ32 i -i i -i ζ6 ζ65 ζ6 ζ3 ζ65 ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 1 1 linear of order 12 ρ18 1 -1 1 -1 ζ32 ζ3 i i -i -i ζ3 ζ6 ζ65 ζ6 ζ32 ζ65 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 1 -1 linear of order 12 ρ19 1 -1 1 -1 ζ3 ζ32 -i -i i i ζ32 ζ65 ζ6 ζ65 ζ3 ζ6 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 1 -1 linear of order 12 ρ20 1 1 -1 -1 ζ32 ζ3 i -i i -i ζ65 ζ6 ζ65 ζ32 ζ6 ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 1 1 linear of order 12 ρ21 1 -1 1 -1 ζ3 ζ32 i i -i -i ζ32 ζ65 ζ6 ζ65 ζ3 ζ6 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 1 -1 linear of order 12 ρ22 1 -1 1 -1 ζ32 ζ3 -i -i i i ζ3 ζ6 ζ65 ζ6 ζ32 ζ65 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 1 -1 linear of order 12 ρ23 1 1 -1 -1 ζ3 ζ32 -i i -i i ζ6 ζ65 ζ6 ζ3 ζ65 ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 1 1 linear of order 12 ρ24 1 1 -1 -1 ζ32 ζ3 -i i -i i ζ65 ζ6 ζ65 ζ32 ζ6 ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 1 1 linear of order 12 ρ25 12 -12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 orthogonal faithful ρ26 12 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from F13

Permutation representations of C2×F13
On 26 points - transitive group 26T10
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)
(2 12 5 6 4 8 13 3 10 9 11 7)(15 25 18 19 17 21 26 16 23 22 24 20)

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

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

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

G:=TransitiveGroup(26,10);

Matrix representation of C2×F13 in GL12(ℤ)

 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1
,
 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0
,
 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0

G:=sub<GL(12,Integers())| [-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1],[-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0],[-1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0] >;

C2×F13 in GAP, Magma, Sage, TeX

C_2\times F_{13}
% in TeX

G:=Group("C2xF13");
// GroupNames label

G:=SmallGroup(312,45);
// by ID

G=gap.SmallGroup(312,45);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,60,4804,464,619]);
// Polycyclic

G:=Group<a,b,c|a^2=b^13=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations

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