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## G = S3×Q8order 48 = 24·3

### Direct product of S3 and Q8

Aliases: S3×Q8, C4.6D6, Dic64C2, C6.7C23, C12.6C22, D6.5C22, Dic3.3C22, C32(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 >

Character table of S3×Q8

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6 12A 12B 12C size 1 1 3 3 2 2 2 2 6 6 6 2 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 1 1 1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 linear of order 2 ρ6 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 linear of order 2 ρ7 1 1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 1 -1 linear of order 2 ρ8 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ9 2 2 0 0 -1 2 2 2 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 0 0 -1 -2 2 -2 0 0 0 -1 1 1 -1 orthogonal lifted from D6 ρ11 2 2 0 0 -1 -2 -2 2 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ12 2 2 0 0 -1 2 -2 -2 0 0 0 -1 1 -1 1 orthogonal lifted from D6 ρ13 2 -2 2 -2 2 0 0 0 0 0 0 -2 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 -2 2 2 0 0 0 0 0 0 -2 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 4 -4 0 0 -2 0 0 0 0 0 0 2 0 0 0 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);

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

 4 2 0 0 2 0 0 0 0 0 4 2 0 0 2 0
,
 4 2 0 0 0 1 0 0 0 0 1 0 0 0 3 4
,
 0 0 1 3 0 0 3 0 0 3 0 0 3 4 0 0
,
 0 0 3 4 0 0 4 0 0 1 0 0 1 3 0 0
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

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