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G = D4.A4order 96 = 25·3

The non-split extension by D4 of A4 acting through Inn(D4)

non-abelian, soluble

Aliases: D4.A4, 2- 1+4⋊C3, D4SL2(𝔽3), SL2(𝔽3).5C22, C4○D4⋊C6, (C2×Q8)⋊C6, C4.A44C2, C4.3(C2×A4), Q8.3(C2×C6), C2.7(C22×A4), C22.5(C2×A4), (C2×SL2(𝔽3))⋊1C2, SmallGroup(96,202)

Series: Derived Chief Lower central Upper central

C1C2Q8 — D4.A4
C1C2Q8SL2(𝔽3)C2×SL2(𝔽3) — D4.A4
Q8 — D4.A4
C1C2D4

Generators and relations for D4.A4
 G = < a,b,c,d,e | a4=b2=e3=1, c2=d2=a2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a2c, ece-1=a2cd, ede-1=c >

2C2
2C2
6C2
4C3
3C4
3C22
3C4
3C4
4C6
8C6
8C6
3Q8
3C2×C4
3Q8
3C2×C4
3C2×C4
3Q8
3C2×C4
3D4
3C2×C4
3D4
3D4
4C2×C6
4C12
4C2×C6
3C4○D4
3C4○D4
3C2×Q8
3C4○D4
4C3×D4

Character table of D4.A4

 class 12A2B2C2D3A3B4A4B4C4D6A6B6C6D6E6F12A12B
 size 1122644266644888888
ρ11111111111111111111    trivial
ρ211-11-111-1-11111-11-11-1-1    linear of order 2
ρ3111-1-111-111-1111-11-1-1-1    linear of order 2
ρ411-1-11111-11-111-1-1-1-111    linear of order 2
ρ511111ζ32ζ31111ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ6111-1-1ζ32ζ3-111-1ζ32ζ3ζ3ζ6ζ32ζ65ζ65ζ6    linear of order 6
ρ711-1-11ζ3ζ321-11-1ζ3ζ32ζ6ζ65ζ65ζ6ζ32ζ3    linear of order 6
ρ811111ζ3ζ321111ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ9111-1-1ζ3ζ32-111-1ζ3ζ32ζ32ζ65ζ3ζ6ζ6ζ65    linear of order 6
ρ1011-11-1ζ3ζ32-1-111ζ3ζ32ζ6ζ3ζ65ζ32ζ6ζ65    linear of order 6
ρ1111-11-1ζ32ζ3-1-111ζ32ζ3ζ65ζ32ζ6ζ3ζ65ζ6    linear of order 6
ρ1211-1-11ζ32ζ31-11-1ζ32ζ3ζ65ζ6ζ6ζ65ζ3ζ32    linear of order 6
ρ13333-3100-3-1-1100000000    orthogonal lifted from C2×A4
ρ143333-1003-1-1-100000000    orthogonal lifted from A4
ρ1533-3-3-10031-1100000000    orthogonal lifted from C2×A4
ρ1633-33100-31-1-100000000    orthogonal lifted from C2×A4
ρ174-4000-2-2000022000000    symplectic faithful, Schur index 2
ρ184-40001+-31--30000-1--3-1+-3000000    complex faithful
ρ194-40001--31+-30000-1+-3-1--3000000    complex faithful

Permutation representations of D4.A4
On 16 points - transitive group 16T180
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 14 11 16)(10 15 12 13)
(1 13 3 15)(2 14 4 16)(5 9 7 11)(6 10 8 12)
(5 14 11)(6 15 12)(7 16 9)(8 13 10)

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

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

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

G:=TransitiveGroup(16,180);

D4.A4 is a maximal subgroup of
D4.S4  D4.3S4  SD16.A4  D8.A4  D4.4S4  D4.5S4  2- 1+43C6  SL2(𝔽3).11D6  D12.A4  D4.A5  SL2(𝔽3).11D10  D20.A4
D4.A4 is a maximal quotient of
(C2×Q8)⋊C12  C4○D4⋊C12  SL2(𝔽3)⋊5D4  D4×SL2(𝔽3)  SL2(𝔽3)⋊3Q8  2- 1+4⋊C9  SL2(𝔽3).11D6  D12.A4  SL2(𝔽3).11D10  D20.A4

Matrix representation of D4.A4 in GL4(𝔽3) generated by

0200
1000
0102
1010
,
0100
1000
0201
1010
,
2010
0101
1010
0102
,
0020
0102
1000
0202
,
1010
0001
0010
0202
G:=sub<GL(4,GF(3))| [0,1,0,1,2,0,1,0,0,0,0,1,0,0,2,0],[0,1,0,1,1,0,2,0,0,0,0,1,0,0,1,0],[2,0,1,0,0,1,0,1,1,0,1,0,0,1,0,2],[0,0,1,0,0,1,0,2,2,0,0,0,0,2,0,2],[1,0,0,0,0,0,0,2,1,0,1,0,0,1,0,2] >;

D4.A4 in GAP, Magma, Sage, TeX

D_4.A_4
% in TeX

G:=Group("D4.A4");
// GroupNames label

G:=SmallGroup(96,202);
// by ID

G=gap.SmallGroup(96,202);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-2,601,159,117,286,202,88]);
// Polycyclic

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

Export

Subgroup lattice of D4.A4 in TeX
Character table of D4.A4 in TeX

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