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G = D4.S4  order 192 = 26·3

2nd non-split extension by D4 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: D4.2S4, U2(𝔽3)⋊3C2, 2- 1+4.S3, SL2(𝔽3).9D4, C4.7(C2×S4), D4.A4.C2, C4○D4.3D6, C4.S4⋊3C2, Q8.6(C3⋊D4), C4.A4.3C22, C2.14(A4⋊D4), SmallGroup(192,989)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — Q8 — C4.A4 — D4.S4
Chief series
C1 — C2 — Q8 — SL2(𝔽3) — C4.A4 — C4.S4 — D4.S4
Lower central
SL2(𝔽3) — C4.A4 — D4.S4
Upper central
C1 — C2 — C4 — D4

Generators and relations for D4.S4
 G = < a,b,c,d,e,f | a4=b2=e3=1, c2=d2=f2=a2, bab=faf-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, fbf-1=ab, dcd-1=a2c, ece-1=a2cd, fcf-1=cd, ede-1=c, fdf-1=a2d, fef-1=e-1 >

Subgroups: 277 in 69 conjugacy classes, 13 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C8, C2×C4, D4, D4, Q8, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C4○D4, C4○D4, C3⋊C8, SL2(𝔽3), Dic6, C3×D4, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, D4.S3, CSU2(𝔽3), C2×SL2(𝔽3), C4.A4, D4.10D4, U2(𝔽3), C4.S4, D4.A4, D4.S4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S4, C2×S4, A4⋊D4, D4.S4

Character table of D4.S4

 class 12A2B2C34A4B4C4D4E4F6A6B6C8A8B12
 size 11468261212122481616242416
ρ111111111111111111    trivial
ρ21111111-1-11-1111-1-11    linear of order 2
ρ311-11111-1-1-111-1-11-11    linear of order 2
ρ411-1111111-1-11-1-1-111    linear of order 2
ρ5220-22-22000020000-2    orthogonal lifted from D4
ρ622-22-12200-20-11100-1    orthogonal lifted from D6
ρ72222-1220020-1-1-100-1    orthogonal lifted from S3
ρ8220-2-1-220000-1√-3-√-3001    complex lifted from C3⋊D4
ρ9220-2-1-220000-1-√-3√-3001    complex lifted from C3⋊D4
ρ1033-3-103-11111000-1-10    orthogonal lifted from C2×S4
ρ1133-3-103-1-1-11-1000110    orthogonal lifted from C2×S4
ρ12333-103-1-1-1-11000-110    orthogonal lifted from S4
ρ13333-103-111-1-10001-10    orthogonal lifted from S4
ρ144-400-2002-200200000    symplectic faithful, Schur index 2
ρ154-400-200-2200200000    symplectic faithful, Schur index 2
ρ1666020-6-20000000000    orthogonal lifted from A4⋊D4
ρ178-8002000000-200000    symplectic faithful, Schur index 2

Smallest permutation representation of D4.S4
►On 32 points
Generators in S32
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)

(1 4)(2 3)(5 6)(7 8)(9 11)(13 16)(14 15)(17 18)(19 20)(22 24)(26 28)(29 31)

(1 8 3 6)(2 5 4 7)(9 28 11 26)(10 25 12 27)(13 18 15 20)(14 19 16 17)(21 30 23 32)(22 31 24 29)

(1 18 3 20)(2 19 4 17)(5 14 7 16)(6 15 8 13)(9 24 11 22)(10 21 12 23)(25 32 27 30)(26 29 28 31)

(5 19 16)(6 20 13)(7 17 14)(8 18 15)(9 28 31)(10 25 32)(11 26 29)(12 27 30)

(1 24 3 22)(2 23 4 21)(5 27 7 25)(6 26 8 28)(9 20 11 18)(10 19 12 17)(13 29 15 31)(14 32 16 30)
 
G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,4)(2,3)(5,6)(7,8)(9,11)(13,16)(14,15)(17,18)(19,20)(22,24)(26,28)(29,31), (1,8,3,6)(2,5,4,7)(9,28,11,26)(10,25,12,27)(13,18,15,20)(14,19,16,17)(21,30,23,32)(22,31,24,29), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,24,11,22)(10,21,12,23)(25,32,27,30)(26,29,28,31), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,28,31)(10,25,32)(11,26,29)(12,27,30), (1,24,3,22)(2,23,4,21)(5,27,7,25)(6,26,8,28)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30)>;
 
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)(25,26,27,28)(29,30,31,32), (1,4)(2,3)(5,6)(7,8)(9,11)(13,16)(14,15)(17,18)(19,20)(22,24)(26,28)(29,31), (1,8,3,6)(2,5,4,7)(9,28,11,26)(10,25,12,27)(13,18,15,20)(14,19,16,17)(21,30,23,32)(22,31,24,29), (1,18,3,20)(2,19,4,17)(5,14,7,16)(6,15,8,13)(9,24,11,22)(10,21,12,23)(25,32,27,30)(26,29,28,31), (5,19,16)(6,20,13)(7,17,14)(8,18,15)(9,28,31)(10,25,32)(11,26,29)(12,27,30), (1,24,3,22)(2,23,4,21)(5,27,7,25)(6,26,8,28)(9,20,11,18)(10,19,12,17)(13,29,15,31)(14,32,16,30) );
 
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),(25,26,27,28),(29,30,31,32)], [(1,4),(2,3),(5,6),(7,8),(9,11),(13,16),(14,15),(17,18),(19,20),(22,24),(26,28),(29,31)], [(1,8,3,6),(2,5,4,7),(9,28,11,26),(10,25,12,27),(13,18,15,20),(14,19,16,17),(21,30,23,32),(22,31,24,29)], [(1,18,3,20),(2,19,4,17),(5,14,7,16),(6,15,8,13),(9,24,11,22),(10,21,12,23),(25,32,27,30),(26,29,28,31)], [(5,19,16),(6,20,13),(7,17,14),(8,18,15),(9,28,31),(10,25,32),(11,26,29),(12,27,30)], [(1,24,3,22),(2,23,4,21),(5,27,7,25),(6,26,8,28),(9,20,11,18),(10,19,12,17),(13,29,15,31),(14,32,16,30)]])
 

Matrix representation of D4.S4 ►in GL4(𝔽3) generated by

1210
1102
2222
2112
,
2201
2120
0122
2021
,
2101
0210
0110
1221
,
2001
2210
1111
1001
,
0012
1121
1222
2201
,
0022
1220
2202
2101
G:=sub<GL(4,GF(3))| [1,1,2,2,2,1,2,1,1,0,2,1,0,2,2,2],[2,2,0,2,2,1,1,0,0,2,2,2,1,0,2,1],[2,0,0,1,1,2,1,2,0,1,1,2,1,0,0,1],[2,2,1,1,0,2,1,0,0,1,1,0,1,0,1,1],[0,1,1,2,0,1,2,2,1,2,2,0,2,1,2,1],[0,1,2,2,0,2,2,1,2,2,0,0,2,0,2,1] >;
 

D4.S4 in GAP, Magma, Sage, TeX 

D_4.S_4
 
% in TeX
 
G:=Group("D4.S4");
 
// GroupNames label
 
G:=SmallGroup(192,989);
 
// by ID
 
G=gap.SmallGroup(192,989);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,85,2102,1059,520,451,1684,655,172,1013,404,285,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^4=b^2=e^3=1,c^2=d^2=f^2=a^2,b*a*b=f*a*f^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=a*b,d*c*d^-1=a^2*c,e*c*e^-1=a^2*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=a^2*d,f*e*f^-1=e^-1>;
 
// generators/relations
 

Export

Character table of D4.S4 in TeX

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