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G = C3×Dic6order 72 = 23·32

Direct product of C3 and Dic6

Aliases: C3×Dic6, C12.1C6, C323Q8, C12.7S3, C6.17D6, Dic3.C6, C3⋊(C3×Q8), C4.(C3×S3), C2.3(S3×C6), C6.1(C2×C6), (C3×C12).2C2, (C3×C6).6C22, (C3×Dic3).2C2, SmallGroup(72,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×Dic6
 Chief series C1 — C3 — C6 — C3×C6 — C3×Dic3 — C3×Dic6
 Lower central C3 — C6 — C3×Dic6
 Upper central C1 — C6 — C12

Generators and relations for C3×Dic6
G = < a,b,c | a3=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C3×Dic6

 class 1 2 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L size 1 1 1 1 2 2 2 2 6 6 1 1 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 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 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ5 1 1 ζ3 ζ32 ζ3 1 ζ32 1 -1 -1 ζ32 ζ3 ζ3 1 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ32 1 ζ3 ζ6 ζ65 ζ65 ζ6 linear of order 6 ρ6 1 1 ζ3 ζ32 ζ3 1 ζ32 -1 -1 1 ζ32 ζ3 ζ3 1 ζ32 -1 ζ65 ζ65 ζ6 ζ6 ζ6 -1 ζ65 ζ32 ζ3 ζ65 ζ6 linear of order 6 ρ7 1 1 ζ32 ζ3 ζ32 1 ζ3 -1 -1 1 ζ3 ζ32 ζ32 1 ζ3 -1 ζ6 ζ6 ζ65 ζ65 ζ65 -1 ζ6 ζ3 ζ32 ζ6 ζ65 linear of order 6 ρ8 1 1 ζ32 ζ3 ζ32 1 ζ3 1 -1 -1 ζ3 ζ32 ζ32 1 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ3 1 ζ32 ζ65 ζ6 ζ6 ζ65 linear of order 6 ρ9 1 1 ζ3 ζ32 ζ3 1 ζ32 -1 1 -1 ζ32 ζ3 ζ3 1 ζ32 -1 ζ65 ζ65 ζ6 ζ6 ζ6 -1 ζ65 ζ6 ζ65 ζ3 ζ32 linear of order 6 ρ10 1 1 ζ32 ζ3 ζ32 1 ζ3 1 1 1 ζ3 ζ32 ζ32 1 ζ3 1 ζ32 ζ32 ζ3 ζ3 ζ3 1 ζ32 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ11 1 1 ζ3 ζ32 ζ3 1 ζ32 1 1 1 ζ32 ζ3 ζ3 1 ζ32 1 ζ3 ζ3 ζ32 ζ32 ζ32 1 ζ3 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ12 1 1 ζ32 ζ3 ζ32 1 ζ3 -1 1 -1 ζ3 ζ32 ζ32 1 ζ3 -1 ζ6 ζ6 ζ65 ζ65 ζ65 -1 ζ6 ζ65 ζ6 ζ32 ζ3 linear of order 6 ρ13 2 2 2 2 -1 -1 -1 -2 0 0 2 2 -1 -1 -1 1 1 1 -2 1 1 1 -2 0 0 0 0 orthogonal lifted from D6 ρ14 2 2 2 2 -1 -1 -1 2 0 0 2 2 -1 -1 -1 -1 -1 -1 2 -1 -1 -1 2 0 0 0 0 orthogonal lifted from S3 ρ15 2 -2 2 2 2 2 2 0 0 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 2 2 -1 -1 -1 0 0 0 -2 -2 1 1 1 √3 √3 -√3 0 √3 -√3 -√3 0 0 0 0 0 symplectic lifted from Dic6, Schur index 2 ρ17 2 -2 2 2 -1 -1 -1 0 0 0 -2 -2 1 1 1 -√3 -√3 √3 0 -√3 √3 √3 0 0 0 0 0 symplectic lifted from Dic6, Schur index 2 ρ18 2 -2 -1+√-3 -1-√-3 -1+√-3 2 -1-√-3 0 0 0 1+√-3 1-√-3 1-√-3 -2 1+√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×Q8 ρ19 2 2 -1+√-3 -1-√-3 ζ65 -1 ζ6 2 0 0 -1-√-3 -1+√-3 ζ65 -1 ζ6 -1 ζ65 ζ65 -1-√-3 ζ6 ζ6 -1 -1+√-3 0 0 0 0 complex lifted from C3×S3 ρ20 2 2 -1+√-3 -1-√-3 ζ65 -1 ζ6 -2 0 0 -1-√-3 -1+√-3 ζ65 -1 ζ6 1 ζ3 ζ3 1+√-3 ζ32 ζ32 1 1-√-3 0 0 0 0 complex lifted from S3×C6 ρ21 2 -2 -1-√-3 -1+√-3 -1-√-3 2 -1+√-3 0 0 0 1-√-3 1+√-3 1+√-3 -2 1-√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×Q8 ρ22 2 -2 -1-√-3 -1+√-3 ζ6 -1 ζ65 0 0 0 1-√-3 1+√-3 ζ32 1 ζ3 -√3 ζ4ζ32+2ζ4 ζ43ζ32+2ζ43 0 ζ43ζ3+2ζ43 ζ4ζ3+2ζ4 √3 0 0 0 0 0 complex faithful ρ23 2 2 -1-√-3 -1+√-3 ζ6 -1 ζ65 2 0 0 -1+√-3 -1-√-3 ζ6 -1 ζ65 -1 ζ6 ζ6 -1+√-3 ζ65 ζ65 -1 -1-√-3 0 0 0 0 complex lifted from C3×S3 ρ24 2 -2 -1-√-3 -1+√-3 ζ6 -1 ζ65 0 0 0 1-√-3 1+√-3 ζ32 1 ζ3 √3 ζ43ζ32+2ζ43 ζ4ζ32+2ζ4 0 ζ4ζ3+2ζ4 ζ43ζ3+2ζ43 -√3 0 0 0 0 0 complex faithful ρ25 2 2 -1-√-3 -1+√-3 ζ6 -1 ζ65 -2 0 0 -1+√-3 -1-√-3 ζ6 -1 ζ65 1 ζ32 ζ32 1-√-3 ζ3 ζ3 1 1+√-3 0 0 0 0 complex lifted from S3×C6 ρ26 2 -2 -1+√-3 -1-√-3 ζ65 -1 ζ6 0 0 0 1+√-3 1-√-3 ζ3 1 ζ32 √3 ζ4ζ3+2ζ4 ζ43ζ3+2ζ43 0 ζ43ζ32+2ζ43 ζ4ζ32+2ζ4 -√3 0 0 0 0 0 complex faithful ρ27 2 -2 -1+√-3 -1-√-3 ζ65 -1 ζ6 0 0 0 1+√-3 1-√-3 ζ3 1 ζ32 -√3 ζ43ζ3+2ζ43 ζ4ζ3+2ζ4 0 ζ4ζ32+2ζ4 ζ43ζ32+2ζ43 √3 0 0 0 0 0 complex faithful

Permutation representations of C3×Dic6
On 24 points - transitive group 24T64
Generators in S24
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 16 7 22)(2 15 8 21)(3 14 9 20)(4 13 10 19)(5 24 11 18)(6 23 12 17)

G:=sub<Sym(24)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17)>;

G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,16,7,22)(2,15,8,21)(3,14,9,20)(4,13,10,19)(5,24,11,18)(6,23,12,17) );

G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,16,7,22),(2,15,8,21),(3,14,9,20),(4,13,10,19),(5,24,11,18),(6,23,12,17)]])

G:=TransitiveGroup(24,64);

C3×Dic6 is a maximal subgroup of
Dic6⋊S3  C325SD16  C322Q16  C323Q16  D12⋊S3  Dic3.D6  D6.6D6  C3×S3×Q8  He33Q8  C36.C6  He34Q8  Dic6.A4
C3×Dic6 is a maximal quotient of
He33Q8  C36.C6

Matrix representation of C3×Dic6 in GL2(𝔽13) generated by

 3 0 0 3
,
 7 0 0 2
,
 0 1 12 0
G:=sub<GL(2,GF(13))| [3,0,0,3],[7,0,0,2],[0,12,1,0] >;

C3×Dic6 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_6
% in TeX

G:=Group("C3xDic6");
// GroupNames label

G:=SmallGroup(72,26);
// by ID

G=gap.SmallGroup(72,26);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-3,60,141,66,1204]);
// Polycyclic

G:=Group<a,b,c|a^3=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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