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G = D4.S3  order 48 = 24·3

The non-split extension by D4 of S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.S3, C4.2D6, C6.8D4, C3⋊2SD16, Dic6⋊2C2, C12.2C22, C3⋊C8⋊2C2, (C3×D4).1C2, C2.5(C3⋊D4), SmallGroup(48,16)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — D4.S3
Chief series
C1 — C3 — C6 — C12 — Dic6 — D4.S3
Lower central
C3 — C6 — C12 — D4.S3
Upper central
C1 — C2 — C4 — D4

Generators and relations for D4.S3
 G = < a,b,c,d | a4=b2=c3=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

C1
C2
4C2
C3
C4
2C22
6C4
C6
4C6
D4
3C8
3Q8
C12
2Dic3
2C2×C6
3SD16
Dic6
C3⋊C8
C3×D4
D4.S3

Character table of D4.S3

 class 12A2B34A4B6A6B6C8A8B12
 size 1142212244664
ρ1111111111111    trivial
ρ211111-1111-1-11    linear of order 2
ρ311-11111-1-1-1-11    linear of order 2
ρ411-111-11-1-1111    linear of order 2
ρ5222-120-1-1-100-1    orthogonal lifted from S3
ρ62202-2020000-2    orthogonal lifted from D4
ρ722-2-120-11100-1    orthogonal lifted from D6
ρ8220-1-20-1-√-3√-3001    complex lifted from C3⋊D4
ρ9220-1-20-1√-3-√-3001    complex lifted from C3⋊D4
ρ102-20200-200√-2-√-20    complex lifted from SD16
ρ112-20200-200-√-2√-20    complex lifted from SD16
ρ124-40-200200000    symplectic faithful, Schur index 2

Permutation representations of D4.S3
►On 24 points - transitive group 24T42
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 4)(2 3)(6 8)(9 11)(13 14)(15 16)(17 20)(18 19)(22 24)

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

(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)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(22,24), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,21,10)(6,22,11)(7,23,12)(8,24,9), (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( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(22,24), (1,19,14)(2,20,15)(3,17,16)(4,18,13)(5,21,10)(6,22,11)(7,23,12)(8,24,9), (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([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4),(2,3),(6,8),(9,11),(13,14),(15,16),(17,20),(18,19),(22,24)], [(1,19,14),(2,20,15),(3,17,16),(4,18,13),(5,21,10),(6,22,11),(7,23,12),(8,24,9)], [(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(24,42);
 

D4.S3 is a maximal subgroup of
 D8⋊S3  D8⋊3S3  S3×SD16  D4.D6  D12⋊6C22  Q8.13D6  Q8.14D6  D4.D9  Dic6⋊S3  D12.S3  C32⋊9SD16  A4⋊SD16  D4.S4  C30.D4  C6.D20  D4.D15  C28.D6  C6.D28  D4.D21  C33⋊7SD16  C33⋊SD16
D4.S3 is a maximal quotient of
 C12.Q8  C6.SD16  D4⋊Dic3  D4.D9  Dic6⋊S3  D12.S3  C32⋊9SD16  A4⋊SD16  C30.D4  C6.D20  D4.D15  C28.D6  C6.D28  D4.D21  C33⋊7SD16  C33⋊SD16

Matrix representation of D4.S3 ►in GL4(𝔽5) generated by

4403
1032
0314
3310
,
4000
1120
0040
3041
,
4010
2444
4000
4120
,
0444
1022
0014
0024
G:=sub<GL(4,GF(5))| [4,1,0,3,4,0,3,3,0,3,1,1,3,2,4,0],[4,1,0,3,0,1,0,0,0,2,4,4,0,0,0,1],[4,2,4,4,0,4,0,1,1,4,0,2,0,4,0,0],[0,1,0,0,4,0,0,0,4,2,1,2,4,2,4,4] >;
 

D4.S3 in GAP, Magma, Sage, TeX 

D_4.S_3
 
% in TeX
 
G:=Group("D4.S3");
 
// GroupNames label
 
G:=SmallGroup(48,16);
 
// by ID
 
G=gap.SmallGroup(48,16);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-3,40,61,182,97,42,804]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^3=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D4.S3 in TeX
Character table of D4.S3 in TeX

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