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## G = C3×C6order 18 = 2·32

### Abelian group of type [3,6]

Aliases: C3×C6, SmallGroup(18,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6
 Chief series C1 — C3 — C32 — C3×C6
 Lower central C1 — C3×C6
 Upper central C1 — C3×C6

Generators and relations for C3×C6
G = < a,b | a3=b6=1, ab=ba >

Character table of C3×C6

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C 6D 6E 6F 6G 6H size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ρ1 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 linear of order 2 ρ3 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 1 ζ32 1 ζ3 ζ3 ζ3 ζ32 ζ32 1 linear of order 3 ρ4 1 -1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 1 ζ6 -1 ζ65 ζ65 ζ65 ζ6 ζ6 -1 linear of order 6 ρ5 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 1 ζ3 1 ζ32 ζ32 ζ32 ζ3 ζ3 1 linear of order 3 ρ6 1 -1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 1 ζ65 -1 ζ6 ζ6 ζ6 ζ65 ζ65 -1 linear of order 6 ρ7 1 1 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ32 1 ζ3 ζ32 1 ζ3 ζ3 linear of order 3 ρ8 1 -1 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ6 ζ6 -1 ζ65 ζ6 -1 ζ65 ζ65 linear of order 6 ρ9 1 1 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 ζ3 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 linear of order 3 ρ10 1 -1 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 ζ3 ζ65 ζ6 ζ65 ζ6 -1 ζ6 -1 ζ65 linear of order 6 ρ11 1 1 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 ζ3 1 ζ32 ζ32 1 ζ3 ζ3 ζ32 ζ3 linear of order 3 ρ12 1 -1 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 ζ3 -1 ζ6 ζ6 -1 ζ65 ζ65 ζ6 ζ65 linear of order 6 ρ13 1 1 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ3 1 ζ32 ζ3 1 ζ32 ζ32 linear of order 3 ρ14 1 -1 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ65 ζ65 -1 ζ6 ζ65 -1 ζ6 ζ6 linear of order 6 ρ15 1 1 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 ζ32 1 ζ3 ζ3 1 ζ32 ζ32 ζ3 ζ32 linear of order 3 ρ16 1 -1 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 ζ32 -1 ζ65 ζ65 -1 ζ6 ζ6 ζ65 ζ6 linear of order 6 ρ17 1 1 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 ζ32 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 linear of order 3 ρ18 1 -1 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 ζ32 ζ6 ζ65 ζ6 ζ65 -1 ζ65 -1 ζ6 linear of order 6

Permutation representations of C3×C6
Regular action on 18 points - transitive group 18T2
Generators in S18
(1 12 15)(2 7 16)(3 8 17)(4 9 18)(5 10 13)(6 11 14)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

G:=sub<Sym(18)| (1,12,15)(2,7,16)(3,8,17)(4,9,18)(5,10,13)(6,11,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;

G:=Group( (1,12,15)(2,7,16)(3,8,17)(4,9,18)(5,10,13)(6,11,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );

G=PermutationGroup([[(1,12,15),(2,7,16),(3,8,17),(4,9,18),(5,10,13),(6,11,14)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)]])

G:=TransitiveGroup(18,2);

C3×C6 is a maximal subgroup of   C3⋊Dic3

Matrix representation of C3×C6 in GL2(𝔽7) generated by

 4 0 0 4
,
 6 0 0 3
G:=sub<GL(2,GF(7))| [4,0,0,4],[6,0,0,3] >;

C3×C6 in GAP, Magma, Sage, TeX

C_3\times C_6
% in TeX

G:=Group("C3xC6");
// GroupNames label

G:=SmallGroup(18,5);
// by ID

G=gap.SmallGroup(18,5);
# by ID

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

G:=Group<a,b|a^3=b^6=1,a*b=b*a>;
// generators/relations

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