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## G = D4.A4order 96 = 25·3

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

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

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

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 >

Character table of D4.A4

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 12A 12B size 1 1 2 2 6 4 4 2 6 6 6 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 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 -1 1 -1 -1 linear of order 2 ρ3 1 1 1 -1 -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 -1 -1 1 1 linear of order 2 ρ5 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ6 1 1 1 -1 -1 ζ32 ζ3 -1 1 1 -1 ζ32 ζ3 ζ3 ζ6 ζ32 ζ65 ζ65 ζ6 linear of order 6 ρ7 1 1 -1 -1 1 ζ3 ζ32 1 -1 1 -1 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 ζ32 ζ3 linear of order 6 ρ8 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ9 1 1 1 -1 -1 ζ3 ζ32 -1 1 1 -1 ζ3 ζ32 ζ32 ζ65 ζ3 ζ6 ζ6 ζ65 linear of order 6 ρ10 1 1 -1 1 -1 ζ3 ζ32 -1 -1 1 1 ζ3 ζ32 ζ6 ζ3 ζ65 ζ32 ζ6 ζ65 linear of order 6 ρ11 1 1 -1 1 -1 ζ32 ζ3 -1 -1 1 1 ζ32 ζ3 ζ65 ζ32 ζ6 ζ3 ζ65 ζ6 linear of order 6 ρ12 1 1 -1 -1 1 ζ32 ζ3 1 -1 1 -1 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 ζ3 ζ32 linear of order 6 ρ13 3 3 3 -3 1 0 0 -3 -1 -1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 3 3 3 -1 0 0 3 -1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ15 3 3 -3 -3 -1 0 0 3 1 -1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ16 3 3 -3 3 1 0 0 -3 1 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ17 4 -4 0 0 0 -2 -2 0 0 0 0 2 2 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ18 4 -4 0 0 0 1+√-3 1-√-3 0 0 0 0 -1-√-3 -1+√-3 0 0 0 0 0 0 complex faithful ρ19 4 -4 0 0 0 1-√-3 1+√-3 0 0 0 0 -1+√-3 -1-√-3 0 0 0 0 0 0 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

 0 2 0 0 1 0 0 0 0 1 0 2 1 0 1 0
,
 0 1 0 0 1 0 0 0 0 2 0 1 1 0 1 0
,
 2 0 1 0 0 1 0 1 1 0 1 0 0 1 0 2
,
 0 0 2 0 0 1 0 2 1 0 0 0 0 2 0 2
,
 1 0 1 0 0 0 0 1 0 0 1 0 0 2 0 2
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

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