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G = D6⋊S3  order 72 = 23·32

1st semidirect product of D6 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: D6⋊1S3, C6.3D6, C32⋊2D4, C2.3S32, (S3×C6)⋊1C2, C3⋊2(C3⋊D4), C3⋊Dic3⋊2C2, (C3×C6).3C22, SmallGroup(72,22)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — D6⋊S3
Chief series
C1 — C3 — C32 — C3×C6 — S3×C6 — D6⋊S3
Lower central
C32 — C3×C6 — D6⋊S3
Upper central
C1 — C2

Generators and relations for D6⋊S3
 G = < a,b,c,d | a6=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=c-1 >

C1
C2
6C2
6C2
C3
C3
2C3
3C22
3C22
9C4
C6
C6
2C6
2S3
2S3
6C6
6C6
C32
9D4
D6
D6
3C2×C6
3C2×C6
3Dic3
3Dic3
6Dic3
C3×C6
2C3×S3
2C3×S3
3C3⋊D4
3C3⋊D4
S3×C6
S3×C6
C3⋊Dic3
D6⋊S3

Character table of D6⋊S3

 class 12A2B2C3A3B3C46A6B6C6D6E6F6G
 size 1166224182246666
ρ1111111111111111    trivial
ρ2111-1111-1111-111-1    linear of order 2
ρ311-1-11111111-1-1-1-1    linear of order 2
ρ411-11111-11111-1-11    linear of order 2
ρ522-202-1-102-1-10110    orthogonal lifted from D6
ρ62-2002220-2-2-20000    orthogonal lifted from D4
ρ722202-1-102-1-10-1-10    orthogonal lifted from S3
ρ82202-12-10-12-1-100-1    orthogonal lifted from S3
ρ9220-2-12-10-12-11001    orthogonal lifted from D6
ρ102-200-12-101-21-√-300√-3    complex lifted from C3⋊D4
ρ112-2002-1-10-2110√-3-√-30    complex lifted from C3⋊D4
ρ122-2002-1-10-2110-√-3√-30    complex lifted from C3⋊D4
ρ132-200-12-101-21√-300-√-3    complex lifted from C3⋊D4
ρ144400-2-210-2-210000    orthogonal lifted from S32
ρ154-400-2-21022-10000    symplectic faithful, Schur index 2

Permutation representations of D6⋊S3
►On 24 points - transitive group 24T61
Generators in S24
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)

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

(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
 
G:=sub<Sym(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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)>;
 
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,21,23)(20,22,24), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) );
 
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,19),(14,24),(15,23),(16,22),(17,21),(18,20)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,21,23),(20,22,24)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)]])
 
G:=TransitiveGroup(24,61);
 

D6⋊S3 is a maximal subgroup of
 C32⋊D8  C32⋊2SD16  D12⋊5S3  D6.D6  D6⋊D6  D6.4D6  S3×C3⋊D4  D6⋊D9  He3⋊3D4  C33⋊6D4  C33⋊9D4  D6.S4  D6⋊S4  D6⋊D15  C32⋊3D20
D6⋊S3 is a maximal quotient of
 C32⋊2D8  Dic6⋊S3  C32⋊2Q16  D6⋊Dic3  C62.C22  D6⋊D9  He3⋊2D4  C33⋊6D4  C33⋊9D4  D6⋊S4  D6⋊D15  C32⋊3D20

Matrix representation of D6⋊S3 ►in GL4(𝔽5) generated by

0010
0104
4010
0100
,
0400
4000
0004
0040
,
0040
0004
1040
0104
,
0200
3000
0203
3020
G:=sub<GL(4,GF(5))| [0,0,4,0,0,1,0,1,1,0,1,0,0,4,0,0],[0,4,0,0,4,0,0,0,0,0,0,4,0,0,4,0],[0,0,1,0,0,0,0,1,4,0,4,0,0,4,0,4],[0,3,0,3,2,0,2,0,0,0,0,2,0,0,3,0] >;
 

D6⋊S3 in GAP, Magma, Sage, TeX 

D_6\rtimes S_3
 
% in TeX
 
G:=Group("D6:S3");
 
// GroupNames label
 
G:=SmallGroup(72,22);
 
// by ID
 
G=gap.SmallGroup(72,22);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-3,-3,61,168,1204]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D6⋊S3 in TeX
Character table of D6⋊S3 in TeX

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