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G = D4.4S4order 192 = 26·3

1st non-split extension by D4 of S4 acting through Inn(D4)

non-abelian, soluble

Aliases: D4.4S4, 2- 1+42S3, GL2(𝔽3)⋊5C22, CSU2(𝔽3)⋊5C22, SL2(𝔽3).8C23, C4○D4⋊D6, (C2×Q8)⋊D6, C4.14(C2×S4), D4.A42C2, C4.3S45C2, C4.6S43C2, C4.A42C22, C22.5(C2×S4), C2.19(C22×S4), Q8.D61C2, Q8.9(C22×S3), (C2×GL2(𝔽3))⋊3C2, (C2×SL2(𝔽3))⋊C22, SmallGroup(192,1485)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — D4.4S4
C1C2Q8SL2(𝔽3)GL2(𝔽3)C2×GL2(𝔽3) — D4.4S4
SL2(𝔽3) — D4.4S4
C1C2D4

Generators and relations for D4.4S4
 G = < a,b,c,d,e,f | a4=b2=e3=f2=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=a2c, ece-1=a2cd, fcf=cd, ede-1=c, fdf=a2d, fef=e-1 >

Subgroups: 615 in 152 conjugacy classes, 27 normal (15 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×4], C22 [×2], C22 [×7], S3 [×4], C6 [×3], C8 [×4], C2×C4 [×7], D4, D4 [×12], Q8, Q8 [×4], C23 [×3], Dic3, C12, D6 [×7], C2×C6 [×2], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×2], C2×Q8, C4○D4, C4○D4 [×7], SL2(𝔽3), C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ 1+4, 2- 1+4, CSU2(𝔽3), GL2(𝔽3), GL2(𝔽3) [×2], C2×SL2(𝔽3) [×2], C4.A4, S3×D4, D4○SD16, C2×GL2(𝔽3) [×2], Q8.D6 [×2], C4.6S4, C4.3S4, D4.A4, D4.4S4
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], S4, C22×S3, C2×S4 [×3], C22×S4, D4.4S4

Character table of D4.4S4

 class 12A2B2C2D2E2F2G34A4B4C4D4E6A6B6C8A8B8C8D8E12
 size 112261212128266612816166612121216
ρ111111111111111111111111    trivial
ρ2111-1-1-1-111-111-1111-1-1-1-111-1    linear of order 2
ρ3111-1-111-11-111-1-111-1111-1-1-1    linear of order 2
ρ411111-1-1-111111-1111-1-1-1-1-11    linear of order 2
ρ511-11-1-1111-1-111-11-1111-11-1-1    linear of order 2
ρ611-1-111-1111-11-1-11-1-1-1-111-11    linear of order 2
ρ711-1-11-11-111-11-111-1-111-1-111    linear of order 2
ρ811-11-11-1-11-1-11111-11-1-11-11-1    linear of order 2
ρ9222-2-2000-1-222-20-1-11000001    orthogonal lifted from D6
ρ1022-2-22000-12-22-20-11100000-1    orthogonal lifted from D6
ρ1122-22-2000-1-2-2220-11-1000001    orthogonal lifted from D6
ρ1222222000-122220-1-1-100000-1    orthogonal lifted from S3
ρ1333-331-1110-31-1-1-1000-1-11-110    orthogonal lifted from C2×S4
ρ1433-3-3-11-11031-11-100011-1-110    orthogonal lifted from C2×S4
ρ15333-3111-10-3-1-11-1000-1-1-1110    orthogonal lifted from C2×S4
ρ163333-1-1-1-103-1-1-1-1000111110    orthogonal lifted from S4
ρ17333-31-1-110-3-1-111000111-1-10    orthogonal lifted from C2×S4
ρ183333-111103-1-1-11000-1-1-1-1-10    orthogonal lifted from S4
ρ1933-3311-1-10-31-1-1100011-11-10    orthogonal lifted from C2×S4
ρ2033-3-3-1-11-1031-111000-1-111-10    orthogonal lifted from C2×S4
ρ214-4000000-2000002002-2-2-20000    complex faithful
ρ224-4000000-200000200-2-22-20000    complex faithful
ρ238-8000000200000-200000000    orthogonal faithful

Permutation representations of D4.4S4
On 16 points - transitive group 16T422
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 4)(5 7)(9 11)(14 16)
(1 6 3 8)(2 7 4 5)(9 16 11 14)(10 13 12 15)
(1 10 3 12)(2 11 4 9)(5 14 7 16)(6 15 8 13)
(5 9 16)(6 10 13)(7 11 14)(8 12 15)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)

G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,4)(5,7)(9,11)(14,16), (1,6,3,8)(2,7,4,5)(9,16,11,14)(10,13,12,15), (1,10,3,12)(2,11,4,9)(5,14,7,16)(6,15,8,13), (5,9,16)(6,10,13)(7,11,14)(8,12,15), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (2,4)(5,7)(9,11)(14,16), (1,6,3,8)(2,7,4,5)(9,16,11,14)(10,13,12,15), (1,10,3,12)(2,11,4,9)(5,14,7,16)(6,15,8,13), (5,9,16)(6,10,13)(7,11,14)(8,12,15), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(2,4),(5,7),(9,11),(14,16)], [(1,6,3,8),(2,7,4,5),(9,16,11,14),(10,13,12,15)], [(1,10,3,12),(2,11,4,9),(5,14,7,16),(6,15,8,13)], [(5,9,16),(6,10,13),(7,11,14),(8,12,15)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15)])

G:=TransitiveGroup(16,422);

Matrix representation of D4.4S4 in GL4(𝔽3) generated by

0020
0022
1000
2100
,
2000
0200
0010
0001
,
1100
1200
0021
0011
,
0100
2000
0011
0012
,
1000
1100
0010
0011
,
2000
0100
0020
0021
G:=sub<GL(4,GF(3))| [0,0,1,2,0,0,0,1,2,2,0,0,0,2,0,0],[2,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,1,2,0,0,0,0,2,1,0,0,1,1],[0,2,0,0,1,0,0,0,0,0,1,1,0,0,1,2],[1,1,0,0,0,1,0,0,0,0,1,1,0,0,0,1],[2,0,0,0,0,1,0,0,0,0,2,2,0,0,0,1] >;

D4.4S4 in GAP, Magma, Sage, TeX

D_4._4S_4
% in TeX

G:=Group("D4.4S4");
// GroupNames label

G:=SmallGroup(192,1485);
// by ID

G=gap.SmallGroup(192,1485);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,1059,451,1684,655,172,1013,404,285,124]);
// Polycyclic

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

Export

Character table of D4.4S4 in TeX

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