Aliases: D4.A4, 2- 1+4⋊C3, D4○SL2(𝔽3), SL2(𝔽3).5C22, C4○D4⋊C6, (C2×Q8)⋊C6, C4.A4⋊4C2, 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
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — D4.A4 |
| Q8 — D4.A4 |
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 |
►On 16 points - transitive group 16T180
(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 14 11 16)(10 15 12 13)
(1 13 3 15)(2 14 4 16)(5 11 7 9)(6 12 8 10)
(5 16 9)(6 13 10)(7 14 11)(8 15 12)
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)(9,14,11,16)(10,15,12,13), (1,13,3,15)(2,14,4,16)(5,11,7,9)(6,12,8,10), (5,16,9)(6,13,10)(7,14,11)(8,15,12)>;
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)(9,14,11,16)(10,15,12,13), (1,13,3,15)(2,14,4,16)(5,11,7,9)(6,12,8,10), (5,16,9)(6,13,10)(7,14,11)(8,15,12) );
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),(9,14,11,16),(10,15,12,13)], [(1,13,3,15),(2,14,4,16),(5,11,7,9),(6,12,8,10)], [(5,16,9),(6,13,10),(7,14,11),(8,15,12)]])
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+4⋊3C6 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
Export
Subgroup lattice of D4.A4 in TeX
Character table of D4.A4 in TeX