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G = C2×C13⋊C3order 78 = 2·3·13

Direct product of C2 and C13⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C13 — C2×C13⋊C3
C1C13C13⋊C3 — C2×C13⋊C3
C13 — C2×C13⋊C3
C1C2

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

13C3
13C6

Character table of C2×C13⋊C3

 class 123A3B6A6B13A13B13C13D26A26B26C26D
 size 111313131333333333
ρ111111111111111    trivial
ρ21-111-1-11111-1-1-1-1    linear of order 2
ρ311ζ32ζ3ζ3ζ3211111111    linear of order 3
ρ41-1ζ3ζ32ζ6ζ651111-1-1-1-1    linear of order 6
ρ51-1ζ32ζ3ζ65ζ61111-1-1-1-1    linear of order 6
ρ611ζ3ζ32ζ32ζ311111111    linear of order 3
ρ73-30000ζ136135132ζ13913313ζ13121310134ζ131113813713121310134131113813713913313136135132    complex faithful
ρ83-30000ζ13121310134ζ136135132ζ1311138137ζ1391331313111381371391331313613513213121310134    complex faithful
ρ93-30000ζ13913313ζ1311138137ζ136135132ζ1312131013413613513213121310134131113813713913313    complex faithful
ρ10330000ζ13913313ζ1311138137ζ136135132ζ13121310134ζ136135132ζ13121310134ζ1311138137ζ13913313    complex lifted from C13⋊C3
ρ11330000ζ13121310134ζ136135132ζ1311138137ζ13913313ζ1311138137ζ13913313ζ136135132ζ13121310134    complex lifted from C13⋊C3
ρ12330000ζ1311138137ζ13121310134ζ13913313ζ136135132ζ13913313ζ136135132ζ13121310134ζ1311138137    complex lifted from C13⋊C3
ρ133-30000ζ1311138137ζ13121310134ζ13913313ζ13613513213913313136135132131213101341311138137    complex faithful
ρ14330000ζ136135132ζ13913313ζ13121310134ζ1311138137ζ13121310134ζ1311138137ζ13913313ζ136135132    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

200
020
002
,
012
022
120
,
112
021
020
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

Export

Subgroup lattice of C2×C13⋊C3 in TeX
Character table of C2×C13⋊C3 in TeX

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