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## G = C2×C13⋊C4order 104 = 23·13

### Direct product of C2 and C13⋊C4

Aliases: C2×C13⋊C4, C26⋊C4, D13⋊C4, D26.C2, D13.C22, C13⋊(C2×C4), SmallGroup(104,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C2×C13⋊C4
 Chief series C1 — C13 — D13 — C13⋊C4 — C2×C13⋊C4
 Lower central C13 — C2×C13⋊C4
 Upper central C1 — C2

Generators and relations for C2×C13⋊C4
G = < a,b,c | a2=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >

Character table of C2×C13⋊C4

 class 1 2A 2B 2C 4A 4B 4C 4D 13A 13B 13C 26A 26B 26C size 1 1 13 13 13 13 13 13 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 -1 -1 -i -i i i 1 1 1 1 1 1 linear of order 4 ρ6 1 1 -1 -1 i i -i -i 1 1 1 1 1 1 linear of order 4 ρ7 1 -1 -1 1 i -i -i i 1 1 1 -1 -1 -1 linear of order 4 ρ8 1 -1 -1 1 -i i i -i 1 1 1 -1 -1 -1 linear of order 4 ρ9 4 4 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 orthogonal lifted from C13⋊C4 ρ10 4 -4 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 orthogonal faithful ρ11 4 4 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 orthogonal lifted from C13⋊C4 ρ12 4 -4 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 orthogonal faithful ρ13 4 4 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 orthogonal lifted from C13⋊C4 ρ14 4 -4 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 orthogonal faithful

Permutation representations of C2×C13⋊C4
On 26 points - transitive group 26T7
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 22 13 19)(3 17 12 24)(4 25 11 16)(5 20 10 21)(6 15 9 26)(7 23 8 18)

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

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

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

G:=TransitiveGroup(26,7);

C2×C13⋊C4 is a maximal subgroup of   C52⋊C4  D13.D4
C2×C13⋊C4 is a maximal quotient of   D13⋊C8  C52.C4  C52⋊C4  C13⋊M4(2)  D13.D4

Matrix representation of C2×C13⋊C4 in GL4(𝔽5) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 3 0 1 0 1 0 0 0 4 0 0 4 2 1 0 0
,
 1 2 3 0 0 2 2 1 0 0 2 0 0 3 3 0
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,1,4,2,0,0,0,1,1,0,0,0,0,0,4,0],[1,0,0,0,2,2,0,3,3,2,2,3,0,1,0,0] >;

C2×C13⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_{13}\rtimes C_4
% in TeX

G:=Group("C2xC13:C4");
// GroupNames label

G:=SmallGroup(104,12);
// by ID

G=gap.SmallGroup(104,12);
# by ID

G:=PCGroup([4,-2,-2,-2,-13,16,1027,395]);
// Polycyclic

G:=Group<a,b,c|a^2=b^13=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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