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## G = S3×Dic3order 72 = 23·32

### Direct product of S3 and Dic3

Aliases: S3×Dic3, D6.S3, C6.1D6, (C3×S3)⋊C4, C2.1S32, C33(C4×S3), (S3×C6).C2, C322(C2×C4), C3⋊Dic31C2, C31(C2×Dic3), (C3×Dic3)⋊2C2, (C3×C6).1C22, SmallGroup(72,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S3×Dic3
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3
 Lower central C32 — S3×Dic3
 Upper central C1 — C2

Generators and relations for S3×Dic3
G = < a,b,c,d | a3=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Character table of S3×Dic3

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 12A 12B size 1 1 3 3 2 2 4 3 3 9 9 2 2 4 6 6 6 6 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 -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 linear of order 2 ρ5 1 -1 -1 1 1 1 1 -i i -i i -1 -1 -1 1 -1 i -i linear of order 4 ρ6 1 -1 1 -1 1 1 1 i -i -i i -1 -1 -1 -1 1 -i i linear of order 4 ρ7 1 -1 1 -1 1 1 1 -i i i -i -1 -1 -1 -1 1 i -i linear of order 4 ρ8 1 -1 -1 1 1 1 1 i -i i -i -1 -1 -1 1 -1 -i i linear of order 4 ρ9 2 2 0 0 2 -1 -1 -2 -2 0 0 2 -1 -1 0 0 1 1 orthogonal lifted from D6 ρ10 2 2 -2 -2 -1 2 -1 0 0 0 0 -1 2 -1 1 1 0 0 orthogonal lifted from D6 ρ11 2 2 0 0 2 -1 -1 2 2 0 0 2 -1 -1 0 0 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 -1 2 -1 0 0 0 0 -1 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ13 2 -2 -2 2 -1 2 -1 0 0 0 0 1 -2 1 -1 1 0 0 symplectic lifted from Dic3, Schur index 2 ρ14 2 -2 2 -2 -1 2 -1 0 0 0 0 1 -2 1 1 -1 0 0 symplectic lifted from Dic3, Schur index 2 ρ15 2 -2 0 0 2 -1 -1 2i -2i 0 0 -2 1 1 0 0 i -i complex lifted from C4×S3 ρ16 2 -2 0 0 2 -1 -1 -2i 2i 0 0 -2 1 1 0 0 -i i complex lifted from C4×S3 ρ17 4 4 0 0 -2 -2 1 0 0 0 0 -2 -2 1 0 0 0 0 orthogonal lifted from S32 ρ18 4 -4 0 0 -2 -2 1 0 0 0 0 2 2 -1 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of S3×Dic3
On 24 points - transitive group 24T60
Generators in S24
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 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 8 4 11)(2 7 5 10)(3 12 6 9)(13 20 16 23)(14 19 17 22)(15 24 18 21)

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

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

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

G:=TransitiveGroup(24,60);

S3×Dic3 is a maximal subgroup of
D125S3  D12⋊S3  C4×S32  D6.3D6  D6.4D6  C6.S32  He3⋊(C2×C4)  C339(C2×C4)  Dic3.4S4  D30.S3
S3×Dic3 is a maximal quotient of
D6.Dic3  D6⋊Dic3  Dic3⋊Dic3  C6.S32  C339(C2×C4)  D30.S3

Matrix representation of S3×Dic3 in GL4(𝔽5) generated by

 0 0 0 4 0 4 3 0 0 3 0 0 1 0 0 4
,
 0 1 0 0 1 0 0 0 0 0 0 2 0 0 3 0
,
 0 0 0 1 0 0 3 0 0 3 1 0 4 0 0 1
,
 0 3 1 0 3 0 0 2 0 0 0 4 0 0 1 0
G:=sub<GL(4,GF(5))| [0,0,0,1,0,4,3,0,0,3,0,0,4,0,0,4],[0,1,0,0,1,0,0,0,0,0,0,3,0,0,2,0],[0,0,0,4,0,0,3,0,0,3,1,0,1,0,0,1],[0,3,0,0,3,0,0,0,1,0,0,1,0,2,4,0] >;

S3×Dic3 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_3
% in TeX

G:=Group("S3xDic3");
// GroupNames label

G:=SmallGroup(72,20);
// by ID

G=gap.SmallGroup(72,20);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,26,168,1204]);
// Polycyclic

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

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