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G = C3×C6order 18 = 2·32

Abelian group of type [3,6]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C6, SmallGroup(18,5)

Series: Derived Chief Lower central Upper central

C1 — C3×C6
C1C3C32 — C3×C6
C1 — C3×C6
C1 — C3×C6

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


Character table of C3×C6

 class 123A3B3C3D3E3F3G3H6A6B6C6D6E6F6G6H
 size 111111111111111111
ρ1111111111111111111    trivial
ρ21-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111ζ3ζ3ζ3ζ32ζ32ζ321ζ321ζ3ζ3ζ3ζ32ζ321    linear of order 3
ρ41-11ζ3ζ3ζ3ζ32ζ32ζ321ζ6-1ζ65ζ65ζ65ζ6ζ6-1    linear of order 6
ρ5111ζ32ζ32ζ32ζ3ζ3ζ31ζ31ζ32ζ32ζ32ζ3ζ31    linear of order 3
ρ61-11ζ32ζ32ζ32ζ3ζ3ζ31ζ65-1ζ6ζ6ζ6ζ65ζ65-1    linear of order 6
ρ711ζ321ζ3ζ321ζ3ζ32ζ3ζ32ζ321ζ3ζ321ζ3ζ3    linear of order 3
ρ81-1ζ321ζ3ζ321ζ3ζ32ζ3ζ6ζ6-1ζ65ζ6-1ζ65ζ65    linear of order 6
ρ911ζ32ζ3ζ321ζ321ζ3ζ3ζ3ζ32ζ3ζ321ζ321ζ3    linear of order 3
ρ101-1ζ32ζ3ζ321ζ321ζ3ζ3ζ65ζ6ζ65ζ6-1ζ6-1ζ65    linear of order 6
ρ1111ζ32ζ321ζ3ζ3ζ321ζ31ζ32ζ321ζ3ζ3ζ32ζ3    linear of order 3
ρ121-1ζ32ζ321ζ3ζ3ζ321ζ3-1ζ6ζ6-1ζ65ζ65ζ6ζ65    linear of order 6
ρ1311ζ31ζ32ζ31ζ32ζ3ζ32ζ3ζ31ζ32ζ31ζ32ζ32    linear of order 3
ρ141-1ζ31ζ32ζ31ζ32ζ3ζ32ζ65ζ65-1ζ6ζ65-1ζ6ζ6    linear of order 6
ρ1511ζ3ζ31ζ32ζ32ζ31ζ321ζ3ζ31ζ32ζ32ζ3ζ32    linear of order 3
ρ161-1ζ3ζ31ζ32ζ32ζ31ζ32-1ζ65ζ65-1ζ6ζ6ζ65ζ6    linear of order 6
ρ1711ζ3ζ32ζ31ζ31ζ32ζ32ζ32ζ3ζ32ζ31ζ31ζ32    linear of order 3
ρ181-1ζ3ζ32ζ31ζ31ζ32ζ32ζ6ζ65ζ6ζ65-1ζ65-1ζ6    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);

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

40
04
,
60
03
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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