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G = Q8⋊2S3order 48 = 24·3

The semidirect product of Q8 and S3 acting via S3/C3=C2

Aliases: Q82S3, C6.9D4, C4.3D6, C33SD16, D12.2C2, C12.3C22, C3⋊C83C2, (C3×Q8)⋊1C2, C2.6(C3⋊D4), SmallGroup(48,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — Q8⋊2S3
 Chief series C1 — C3 — C6 — C12 — D12 — Q8⋊2S3
 Lower central C3 — C6 — C12 — Q8⋊2S3
 Upper central C1 — C2 — C4 — Q8

Generators and relations for Q82S3
G = < a,b,c,d | a4=c3=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

Character table of Q82S3

 class 1 2A 2B 3 4A 4B 6 8A 8B 12A 12B 12C size 1 1 12 2 2 4 2 6 6 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 0 -1 2 -2 -1 0 0 -1 1 1 orthogonal lifted from D6 ρ6 2 2 0 -1 2 2 -1 0 0 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 0 2 -2 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ8 2 2 0 -1 -2 0 -1 0 0 1 -√-3 √-3 complex lifted from C3⋊D4 ρ9 2 2 0 -1 -2 0 -1 0 0 1 √-3 -√-3 complex lifted from C3⋊D4 ρ10 2 -2 0 2 0 0 -2 -√-2 √-2 0 0 0 complex lifted from SD16 ρ11 2 -2 0 2 0 0 -2 √-2 -√-2 0 0 0 complex lifted from SD16 ρ12 4 -4 0 -2 0 0 2 0 0 0 0 0 orthogonal faithful

Permutation representations of Q82S3
On 24 points - transitive group 24T36
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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 21)(6 11 22)(7 12 23)(8 9 24)
(2 4)(5 9)(6 12)(7 11)(8 10)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)

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

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

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

G:=TransitiveGroup(24,36);

Matrix representation of Q82S3 in GL4(ℤ) generated by

 0 0 1 0 0 0 0 1 -1 0 0 0 0 -1 0 0
,
 -1 1 0 -1 -1 0 1 -1 0 -1 1 -1 1 -1 1 0
,
 0 -1 0 0 1 -1 0 0 0 0 0 -1 0 0 1 -1
,
 0 1 0 0 1 0 0 0 0 0 0 -1 0 0 -1 0
G:=sub<GL(4,Integers())| [0,0,-1,0,0,0,0,-1,1,0,0,0,0,1,0,0],[-1,-1,0,1,1,0,-1,-1,0,1,1,1,-1,-1,-1,0],[0,1,0,0,-1,-1,0,0,0,0,0,1,0,0,-1,-1],[0,1,0,0,1,0,0,0,0,0,0,-1,0,0,-1,0] >;

Q82S3 in GAP, Magma, Sage, TeX

Q_8\rtimes_2S_3
% in TeX

G:=Group("Q8:2S3");
// GroupNames label

G:=SmallGroup(48,17);
// by ID

G=gap.SmallGroup(48,17);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,61,46,182,97,42,804]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^3=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations

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