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G = Q83S3order 48 = 24·3

The semidirect product of Q8 and S3 acting through Inn(Q8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q83S3, Q8Dic3, C4.7D6, D124C2, C6.8C23, C12.7C22, D6.3C22, Dic3.5C22, (C4×S3)⋊3C2, C33(C4○D4), (C3×Q8)⋊3C2, C2.9(C22×S3), SmallGroup(48,41)

Series: Derived Chief Lower central Upper central

C1C6 — Q83S3
C1C3C6D6C4×S3 — Q83S3
C3C6 — Q83S3
C1C2Q8

Generators and relations for Q83S3
 G = < a,b,c,d | a4=c3=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

6C2
6C2
6C2
3C22
3C22
3C22
3C4
2S3
2S3
2S3
3C2×C4
3D4
3D4
3C2×C4
3D4
3C2×C4
3C4○D4

Character table of Q83S3

 class 12A2B2C2D34A4B4C4D4E612A12B12C
 size 116662222332444
ρ1111111111111111    trivial
ρ2111-111-11-1-1-111-1-1    linear of order 2
ρ311-1-111-1-11111-11-1    linear of order 2
ρ411-11111-1-1-1-11-1-11    linear of order 2
ρ511-11-11-11-11111-1-1    linear of order 2
ρ611-1-1-11111-1-11111    linear of order 2
ρ7111-1-111-1-1111-1-11    linear of order 2
ρ81111-11-1-11-1-11-11-1    linear of order 2
ρ922000-122200-1-1-1-1    orthogonal lifted from S3
ρ1022000-1-22-200-1-111    orthogonal lifted from D6
ρ1122000-12-2-200-111-1    orthogonal lifted from D6
ρ1222000-1-2-2200-11-11    orthogonal lifted from D6
ρ132-200020002i-2i-2000    complex lifted from C4○D4
ρ142-20002000-2i2i-2000    complex lifted from C4○D4
ρ154-4000-2000002000    orthogonal faithful, Schur index 2

Permutation representations of Q83S3
On 24 points - transitive group 24T28
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)
(1 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 19)(14 18)(15 17)(16 20)(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), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,19)(14,18)(15,17)(16,20)(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), (1,4)(2,3)(5,9)(6,12)(7,11)(8,10)(13,19)(14,18)(15,17)(16,20)(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)], [(1,4),(2,3),(5,9),(6,12),(7,11),(8,10),(13,19),(14,18),(15,17),(16,20),(21,24),(22,23)])

G:=TransitiveGroup(24,28);

Matrix representation of Q83S3 in GL4(𝔽5) generated by

3033
4222
2230
3342
,
3320
3213
3033
4232
,
3303
2224
0312
2432
,
0021
0330
0420
1210
G:=sub<GL(4,GF(5))| [3,4,2,3,0,2,2,3,3,2,3,4,3,2,0,2],[3,3,3,4,3,2,0,2,2,1,3,3,0,3,3,2],[3,2,0,2,3,2,3,4,0,2,1,3,3,4,2,2],[0,0,0,1,0,3,4,2,2,3,2,1,1,0,0,0] >;

Q83S3 in GAP, Magma, Sage, TeX

Q_8\rtimes_3S_3
% in TeX

G:=Group("Q8:3S3");
// GroupNames label

G:=SmallGroup(48,41);
// by ID

G=gap.SmallGroup(48,41);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,46,182,97,42,804]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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