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G = D4.4S4  order 192 = 26·3

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

non-abelian, soluble

Aliases: D4.4S4, 2- 1+4⋊2S3, GL2(𝔽3)⋊5C22, CSU2(𝔽3)⋊5C22, SL2(𝔽3).8C23, C4○D4⋊D6, (C2×Q8)⋊D6, C4.14(C2×S4), D4.A4⋊2C2, C4.3S4⋊5C2, C4.6S4⋊3C2, C4.A4⋊2C22, C22.5(C2×S4), C2.19(C22×S4), Q8.D6⋊1C2, Q8.9(C22×S3), (C2×GL2(𝔽3))⋊3C2, (C2×SL2(𝔽3))⋊C22, SmallGroup(192,1485)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — Q8 — SL2(𝔽3) — D4.4S4
Chief series
C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3) — C2×GL2(𝔽3) — D4.4S4
Lower central
SL2(𝔽3) — D4.4S4
Upper central
C1 — C2 — D4

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, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, CSU2(𝔽3), GL2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), C4.A4, S3×D4, D4○SD16, C2×GL2(𝔽3), Q8.D6, C4.6S4, C4.3S4, D4.A4, D4.4S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, C2×S4, 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 8 3 6)(2 5 4 7)(9 16 11 14)(10 13 12 15)

(1 10 3 12)(2 11 4 9)(5 16 7 14)(6 13 8 15)

(5 11 14)(6 12 15)(7 9 16)(8 10 13)

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