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G = Q8⋊2S3  order 48 = 24·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8⋊2S3, C6.9D4, C4.3D6, C3⋊3SD16, D12.2C2, C12.3C22, C3⋊C8⋊3C2, (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 Q8⋊2S3
 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 >

C1
C2
12C2
C3
C4
2C4
6C22
C6
4S3
Q8
3C8
3D4
C12
2D6
2C12
3SD16
D12
C3⋊C8
C3×Q8
Q8⋊2S3

Character table of Q8⋊2S3

 class 12A2B34A4B68A8B12A12B12C
 size 1112224266444
ρ1111111111111    trivial
ρ211111-11-1-11-1-1    linear of order 2
ρ311-11111-1-1111    linear of order 2
ρ411-111-11111-1-1    linear of order 2
ρ5220-12-2-100-111    orthogonal lifted from D6
ρ6220-122-100-1-1-1    orthogonal lifted from S3
ρ72202-20200-200    orthogonal lifted from D4
ρ8220-1-20-1001-√-3√-3    complex lifted from C3⋊D4
ρ9220-1-20-1001√-3-√-3    complex lifted from C3⋊D4
ρ102-20200-2-√-2√-2000    complex lifted from SD16
ρ112-20200-2√-2-√-2000    complex lifted from SD16
ρ124-40-200200000    orthogonal faithful

Permutation representations of Q8⋊2S3
►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);
 

Q8⋊2S3 is a maximal subgroup of
 S3×SD16  Q8⋊3D6  Q16⋊S3  D24⋊C2  Q8.11D6  D4⋊D6  Q8.13D6  Q8⋊2D9  Q8⋊D9  Dic6⋊S3  C32⋊5SD16  C32⋊11SD16  C6.6S4  Q8⋊3S4  Q8.5S4  C20.D6  C15⋊SD16  Q8⋊2D15  C42.D4  C21⋊SD16  Q8⋊2D21  He3⋊SD16  C33⋊6SD16  C33⋊3SD16
Q8⋊2S3 is a maximal quotient of
 C12.Q8  C6.D8  Q8⋊2Dic3  Q8⋊2D9  Dic6⋊S3  C32⋊5SD16  C32⋊11SD16  Q8⋊3S4  C20.D6  C15⋊SD16  Q8⋊2D15  C42.D4  C21⋊SD16  Q8⋊2D21  C33⋊6SD16  C33⋊3SD16

Matrix representation of Q8⋊2S3 ►in GL4(ℤ) generated by

0010
0001
-1000
0-100
,
-110-1
-101-1
0-11-1
1-110
,
0-100
1-100
000-1
001-1
,
0100
1000
000-1
00-10
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] >;
 

Q8⋊2S3 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
 

Export

Subgroup lattice of Q8⋊2S3 in TeX
Character table of Q8⋊2S3 in TeX

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