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G = D4⋊S3order 48 = 24·3

The semidirect product of D4 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4⋊S3, C32D8, C4.1D6, C6.7D4, D122C2, C12.1C22, C3⋊C81C2, (C3×D4)⋊1C2, C2.4(C3⋊D4), SmallGroup(48,15)

Series: Derived Chief Lower central Upper central

C1C12 — D4⋊S3
C1C3C6C12D12 — D4⋊S3
C3C6C12 — D4⋊S3
C1C2C4D4

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

4C2
12C2
2C22
6C22
4S3
4C6
3C8
3D4
2D6
2C2×C6
3D8

Character table of D4⋊S3

 class 12A2B2C346A6B6C8A8B12
 size 1141222244664
ρ1111111111111    trivial
ρ211-1-1111-1-1111    linear of order 2
ρ3111-111111-1-11    linear of order 2
ρ411-11111-1-1-1-11    linear of order 2
ρ52220-12-1-1-100-1    orthogonal lifted from S3
ρ622-20-12-11100-1    orthogonal lifted from D6
ρ722002-220000-2    orthogonal lifted from D4
ρ82-20020-200-220    orthogonal lifted from D8
ρ92-20020-2002-20    orthogonal lifted from D8
ρ102200-1-2-1--3-3001    complex lifted from C3⋊D4
ρ112200-1-2-1-3--3001    complex lifted from C3⋊D4
ρ124-400-20200000    orthogonal faithful, Schur index 2

Permutation representations of D4⋊S3
On 24 points - transitive group 24T37
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 21)(2 24)(3 23)(4 22)(5 19)(6 18)(7 17)(8 20)(9 15)(10 14)(11 13)(12 16)
(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), (1,21)(2,24)(3,23)(4,22)(5,19)(6,18)(7,17)(8,20)(9,15)(10,14)(11,13)(12,16), (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), (1,21)(2,24)(3,23)(4,22)(5,19)(6,18)(7,17)(8,20)(9,15)(10,14)(11,13)(12,16), (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)], [(1,21),(2,24),(3,23),(4,22),(5,19),(6,18),(7,17),(8,20),(9,15),(10,14),(11,13),(12,16)], [(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(24,37);

On 24 points - transitive group 24T43
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 22)(2 21)(3 24)(4 23)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(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), (1,22)(2,21)(3,24)(4,23)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(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), (1,22)(2,21)(3,24)(4,23)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(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)], [(1,22),(2,21),(3,24),(4,23),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13)])

G:=TransitiveGroup(24,43);

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

0024
3442
2110
0220
,
2113
0031
1004
2103
,
0304
0010
0440
1334
,
1004
0010
0100
0004
G:=sub<GL(4,GF(5))| [0,3,2,0,0,4,1,2,2,4,1,2,4,2,0,0],[2,0,1,2,1,0,0,1,1,3,0,0,3,1,4,3],[0,0,0,1,3,0,4,3,0,1,4,3,4,0,0,4],[1,0,0,0,0,0,1,0,0,1,0,0,4,0,0,4] >;

D4⋊S3 in GAP, Magma, Sage, TeX

D_4\rtimes S_3
% in TeX

G:=Group("D4:S3");
// GroupNames label

G:=SmallGroup(48,15);
// by ID

G=gap.SmallGroup(48,15);
# by ID

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

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

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