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## G = D4⋊2S3order 48 = 24·3

### The semidirect product of D4 and S3 acting through Inn(D4)

Aliases: D42S3, D4Dic3, C4.5D6, Dic63C2, C6.6C23, C22.1D6, C12.5C22, D6.2C22, Dic3.4C22, (C4×S3)⋊2C2, (C3×D4)⋊3C2, C32(C4○D4), C3⋊D42C2, (C2×C6).C22, (C2×Dic3)⋊3C2, C2.7(C22×S3), SmallGroup(48,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D4⋊2S3
 Chief series C1 — C3 — C6 — D6 — C4×S3 — D4⋊2S3
 Lower central C3 — C6 — D4⋊2S3
 Upper central C1 — C2 — D4

Generators and relations for D42S3
G = < a,b,c,d | a4=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

Character table of D42S3

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 6A 6B 6C 12 size 1 1 2 2 6 2 2 3 3 6 6 2 4 4 4 ρ1 1 1 1 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 -1 1 -1 linear of order 2 ρ3 1 1 1 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 -1 1 -1 linear of order 2 ρ5 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 linear of order 2 ρ6 1 1 1 -1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 1 linear of order 2 ρ8 1 1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 linear of order 2 ρ9 2 2 -2 -2 0 -1 2 0 0 0 0 -1 1 1 -1 orthogonal lifted from D6 ρ10 2 2 2 -2 0 -1 -2 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 -2 2 0 -1 -2 0 0 0 0 -1 1 -1 1 orthogonal lifted from D6 ρ12 2 2 2 2 0 -1 2 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 -2 0 0 0 2 0 2i -2i 0 0 -2 0 0 0 complex lifted from C4○D4 ρ14 2 -2 0 0 0 2 0 -2i 2i 0 0 -2 0 0 0 complex lifted from C4○D4 ρ15 4 -4 0 0 0 -2 0 0 0 0 0 2 0 0 0 symplectic faithful, Schur index 2

Permutation representations of D42S3
On 24 points - transitive group 24T18
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)(5 8)(6 7)(9 10)(11 12)(13 14)(15 16)(17 20)(18 19)(21 24)(22 23)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)

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

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

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

G:=TransitiveGroup(24,18);

On 24 points - transitive group 24T23
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 20)(2 19)(3 18)(4 17)(5 14)(6 13)(7 16)(8 15)(9 24)(10 23)(11 22)(12 21)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 19 23)(14 20 24)(15 17 21)(16 18 22)
(5 9)(6 10)(7 11)(8 12)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)

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

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

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

G:=TransitiveGroup(24,23);

Matrix representation of D42S3 in GL4(𝔽5) generated by

 0 1 3 0 1 3 2 3 1 4 1 4 0 2 3 1
,
 1 1 0 0 0 4 0 0 3 4 0 4 3 4 4 0
,
 3 0 0 4 2 4 1 1 1 4 0 1 3 0 0 1
,
 1 0 0 1 0 0 1 4 0 1 0 4 0 0 0 4
G:=sub<GL(4,GF(5))| [0,1,1,0,1,3,4,2,3,2,1,3,0,3,4,1],[1,0,3,3,1,4,4,4,0,0,0,4,0,0,4,0],[3,2,1,3,0,4,4,0,0,1,0,0,4,1,1,1],[1,0,0,0,0,0,1,0,0,1,0,0,1,4,4,4] >;

D42S3 in GAP, Magma, Sage, TeX

D_4\rtimes_2S_3
% in TeX

G:=Group("D4:2S3");
// GroupNames label

G:=SmallGroup(48,39);
// by ID

G=gap.SmallGroup(48,39);
# by ID

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

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

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