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## G = C3×Dic3order 36 = 22·32

### Direct product of C3 and Dic3

Aliases: C3×Dic3, C3⋊C12, C6.C6, C6.4S3, C322C4, C2.(C3×S3), (C3×C6).1C2, SmallGroup(36,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×Dic3
 Chief series C1 — C3 — C6 — C3×C6 — C3×Dic3
 Lower central C3 — C3×Dic3
 Upper central C1 — C6

Generators and relations for C3×Dic3
G = < a,b,c | a3=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C3×Dic3

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 12A 12B 12C 12D size 1 1 1 1 2 2 2 3 3 1 1 2 2 2 3 3 3 3 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 ζ3 ζ32 ζ32 1 ζ3 -1 -1 ζ32 ζ3 ζ3 1 ζ32 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ4 1 1 ζ32 ζ3 ζ3 1 ζ32 -1 -1 ζ3 ζ32 ζ32 1 ζ3 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ5 1 1 ζ3 ζ32 ζ32 1 ζ3 1 1 ζ32 ζ3 ζ3 1 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ6 1 1 ζ32 ζ3 ζ3 1 ζ32 1 1 ζ3 ζ32 ζ32 1 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ7 1 -1 1 1 1 1 1 i -i -1 -1 -1 -1 -1 -i i -i i linear of order 4 ρ8 1 -1 1 1 1 1 1 -i i -1 -1 -1 -1 -1 i -i i -i linear of order 4 ρ9 1 -1 ζ3 ζ32 ζ32 1 ζ3 i -i ζ6 ζ65 ζ65 -1 ζ6 ζ43ζ3 ζ4ζ3 ζ43ζ32 ζ4ζ32 linear of order 12 ρ10 1 -1 ζ3 ζ32 ζ32 1 ζ3 -i i ζ6 ζ65 ζ65 -1 ζ6 ζ4ζ3 ζ43ζ3 ζ4ζ32 ζ43ζ32 linear of order 12 ρ11 1 -1 ζ32 ζ3 ζ3 1 ζ32 -i i ζ65 ζ6 ζ6 -1 ζ65 ζ4ζ32 ζ43ζ32 ζ4ζ3 ζ43ζ3 linear of order 12 ρ12 1 -1 ζ32 ζ3 ζ3 1 ζ32 i -i ζ65 ζ6 ζ6 -1 ζ65 ζ43ζ32 ζ4ζ32 ζ43ζ3 ζ4ζ3 linear of order 12 ρ13 2 2 2 2 -1 -1 -1 0 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ14 2 -2 2 2 -1 -1 -1 0 0 -2 -2 1 1 1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 1+√-3 1-√-3 ζ3 1 ζ32 0 0 0 0 complex faithful ρ16 2 -2 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 1-√-3 1+√-3 ζ32 1 ζ3 0 0 0 0 complex faithful ρ17 2 2 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 0 0 complex lifted from C3×S3 ρ18 2 2 -1-√-3 -1+√-3 ζ65 -1 ζ6 0 0 -1+√-3 -1-√-3 ζ6 -1 ζ65 0 0 0 0 complex lifted from C3×S3

Permutation representations of C3×Dic3
On 12 points - transitive group 12T19
Generators in S12
(1 3 5)(2 4 6)(7 11 9)(8 12 10)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 10 4 7)(2 9 5 12)(3 8 6 11)

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

G:=Group( (1,3,5)(2,4,6)(7,11,9)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10,4,7)(2,9,5,12)(3,8,6,11) );

G=PermutationGroup([[(1,3,5),(2,4,6),(7,11,9),(8,12,10)], [(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,10,4,7),(2,9,5,12),(3,8,6,11)]])

G:=TransitiveGroup(12,19);

C3×Dic3 is a maximal subgroup of
C6.D6  C3⋊D12  C322Q8  S3×C12  C32⋊C12  C9⋊C12  He33C4  Dic3.A4  C6.F7  He34Dic3  C393C12  C3⋊F13
C3×Dic3 is a maximal quotient of
C32⋊C12  C9⋊C12  C6.F7  C393C12  C3⋊F13

Polynomial with Galois group C3×Dic3 over ℚ
actionf(x)Disc(f)
12T19x12-4x11-15x10+68x9+62x8-362x7-107x6+798x5+112x4-654x3-63x2+84x+1216·59·194·5412·1019992

Matrix representation of C3×Dic3 in GL2(𝔽7) generated by

 2 0 0 2
,
 5 0 0 3
,
 0 6 1 0
G:=sub<GL(2,GF(7))| [2,0,0,2],[5,0,0,3],[0,1,6,0] >;

C3×Dic3 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_3
% in TeX

G:=Group("C3xDic3");
// GroupNames label

G:=SmallGroup(36,6);
// by ID

G=gap.SmallGroup(36,6);
# by ID

G:=PCGroup([4,-2,-3,-2,-3,24,387]);
// Polycyclic

G:=Group<a,b,c|a^3=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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