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G = S3×Q8  order 48 = 24·3

Direct product of S3 and Q8

direct product, metabelian, supersoluble, monomial, rational, 2-hyperelementary

Aliases: S3×Q8, C4.6D6, Dic6⋊4C2, C6.7C23, C12.6C22, D6.5C22, Dic3.3C22, C3⋊2(C2×Q8), (C3×Q8)⋊2C2, (C4×S3).1C2, C2.8(C22×S3), SmallGroup(48,40)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — S3×Q8
Chief series
C1 — C3 — C6 — D6 — C4×S3 — S3×Q8
Lower central
C3 — C6 — S3×Q8
Upper central
C1 — C2 — Q8

Generators and relations for S3×Q8
 G = < a,b,c,d | a3=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

C1
C2
3C2
3C2
C3
C4
C4
C4
3C4
3C4
3C22
3C4
S3
C6
S3
Q8
3C2×C4
3C2×C4
3Q8
3Q8
3C2×C4
3Q8
Dic3
C12
Dic3
D6
C12
C12
Dic3
3C2×Q8
Dic6
C4×S3
Dic6
C4×S3
Dic6
C3×Q8
C4×S3
S3×Q8

Character table of S3×Q8

 class 12A2B2C34A4B4C4D4E4F612A12B12C
 size 113322226662444
ρ1111111111111111    trivial
ρ211-1-11-11-11-111-1-11    linear of order 2
ρ311-1-11-1-1111-111-1-1    linear of order 2
ρ4111111-1-11-1-11-11-1    linear of order 2
ρ511111-11-1-11-11-1-11    linear of order 2
ρ611-1-11111-1-1-11111    linear of order 2
ρ711-1-111-1-1-1111-11-1    linear of order 2
ρ811111-1-11-1-1111-1-1    linear of order 2
ρ92200-1222000-1-1-1-1    orthogonal lifted from S3
ρ102200-1-22-2000-111-1    orthogonal lifted from D6
ρ112200-1-2-22000-1-111    orthogonal lifted from D6
ρ122200-12-2-2000-11-11    orthogonal lifted from D6
ρ132-22-22000000-2000    symplectic lifted from Q8, Schur index 2
ρ142-2-222000000-2000    symplectic lifted from Q8, Schur index 2
ρ154-400-20000002000    symplectic faithful, Schur index 2

Permutation representations of S3×Q8
►On 24 points - transitive group 24T26
Generators in S24
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)

(5 10)(6 11)(7 12)(8 9)(13 18)(14 19)(15 20)(16 17)

(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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
 
G:=sub<Sym(24)| (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (5,10)(6,11)(7,12)(8,9)(13,18)(14,19)(15,20)(16,17), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)>;
 
G:=Group( (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (5,10)(6,11)(7,12)(8,9)(13,18)(14,19)(15,20)(16,17), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13) );
 
G=PermutationGroup([[(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(5,10),(6,11),(7,12),(8,9),(13,18),(14,19),(15,20),(16,17)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)]])
 
G:=TransitiveGroup(24,26);
 

S3×Q8 is a maximal subgroup of
 D4.D6  Q16⋊S3  Q8.15D6  Q8○D12  Dic3.D6  Q8.6S4  D15⋊Q8  D21⋊Q8  A5⋊Q8  C4.S5
S3×Q8 is a maximal quotient of
 Dic6⋊C4  C12⋊Q8  Dic3.Q8  D6⋊Q8  C4.D12  Dic3⋊Q8  D6⋊3Q8  Dic3.D6  D15⋊Q8  D21⋊Q8

Matrix representation of S3×Q8 ►in GL4(𝔽5) generated by

4200
2000
0042
0020
,
4200
0100
0010
0034
,
0013
0030
0300
3400
,
0034
0040
0100
1300
G:=sub<GL(4,GF(5))| [4,2,0,0,2,0,0,0,0,0,4,2,0,0,2,0],[4,0,0,0,2,1,0,0,0,0,1,3,0,0,0,4],[0,0,0,3,0,0,3,4,1,3,0,0,3,0,0,0],[0,0,0,1,0,0,1,3,3,4,0,0,4,0,0,0] >;
 

S3×Q8 in GAP, Magma, Sage, TeX 

S_3\times Q_8
 
% in TeX
 
G:=Group("S3xQ8");
 
// GroupNames label
 
G:=SmallGroup(48,40);
 
// by ID
 
G=gap.SmallGroup(48,40);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-3,46,97,42,804]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^4=1,d^2=c^2,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
 

Export

Subgroup lattice of S3×Q8 in TeX
Character table of S3×Q8 in TeX

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