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G = C2×C13⋊C3order 78 = 2·3·13

Direct product of C2 and C13⋊C3

Aliases: C2×C13⋊C3, C26⋊C3, C132C6, SmallGroup(78,2)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C13⋊C3
G = < a,b,c | a2=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >

Character table of C2×C13⋊C3

 class 1 2 3A 3B 6A 6B 13A 13B 13C 13D 26A 26B 26C 26D size 1 1 13 13 13 13 3 3 3 3 3 3 3 3 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 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ4 1 -1 ζ3 ζ32 ζ6 ζ65 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ5 1 -1 ζ32 ζ3 ζ65 ζ6 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ6 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ7 3 -3 0 0 0 0 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 complex faithful ρ8 3 -3 0 0 0 0 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 complex faithful ρ9 3 -3 0 0 0 0 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 -ζ139-ζ133-ζ13 complex faithful ρ10 3 3 0 0 0 0 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 complex lifted from C13⋊C3 ρ11 3 3 0 0 0 0 ζ1312+ζ1310+ζ134 ζ136+ζ135+ζ132 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 complex lifted from C13⋊C3 ρ12 3 3 0 0 0 0 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 complex lifted from C13⋊C3 ρ13 3 -3 0 0 0 0 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 -ζ139-ζ133-ζ13 -ζ136-ζ135-ζ132 -ζ1312-ζ1310-ζ134 -ζ1311-ζ138-ζ137 complex faithful ρ14 3 3 0 0 0 0 ζ136+ζ135+ζ132 ζ139+ζ133+ζ13 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ1312+ζ1310+ζ134 ζ1311+ζ138+ζ137 ζ139+ζ133+ζ13 ζ136+ζ135+ζ132 complex lifted from C13⋊C3

Permutation representations of C2×C13⋊C3
On 26 points - transitive group 26T5
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 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)

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

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

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

G:=TransitiveGroup(26,5);

C2×C13⋊C3 is a maximal subgroup of   C26.C6  C26.A4

Matrix representation of C2×C13⋊C3 in GL3(𝔽3) generated by

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

C2×C13⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{13}\rtimes C_3
% in TeX

G:=Group("C2xC13:C3");
// GroupNames label

G:=SmallGroup(78,2);
// by ID

G=gap.SmallGroup(78,2);
# by ID

G:=PCGroup([3,-2,-3,-13,86]);
// Polycyclic

G:=Group<a,b,c|a^2=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations

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