Aliases: Q8.A4, Q8○SL2(𝔽3), 2+ 1+4⋊1C3, SL2(𝔽3)⋊4C22, C4○D4.C6, C4.A4⋊3C2, C4.2(C2×A4), Q8.2(C2×C6), C2.6(C22×A4), SmallGroup(96,201)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — Q8.A4 |
Generators and relations for Q8.A4
G = < a,b,c,d,e | a4=e3=1, b2=c2=d2=a2, bab-1=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 Q8.A4
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 6 | 6 | 6 | 4 | 4 | 2 | 2 | 2 | 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 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | -1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ32 | ζ6 | linear of order 6 |
| ρ9 | 1 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ65 | ζ3 | ζ6 | ζ65 | ζ6 | ζ32 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | -1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ3 | ζ65 | linear of order 6 |
| ρ11 | 1 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ6 | ζ32 | ζ65 | ζ6 | ζ65 | ζ3 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | linear of order 6 |
| ρ13 | 3 | 3 | 1 | -1 | 1 | 0 | 0 | 3 | -3 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | -1 | 1 | 1 | 0 | 0 | -3 | -3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ15 | 3 | 3 | -1 | -1 | -1 | 0 | 0 | 3 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ16 | 3 | 3 | 1 | 1 | -1 | 0 | 0 | -3 | 3 | -3 | -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 | orthogonal faithful |
| 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 24 points - transitive group 24T137
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 11 3 9)(2 10 4 12)(5 23 7 21)(6 22 8 24)(13 18 15 20)(14 17 16 19)
(1 4 3 2)(5 23 7 21)(6 24 8 22)(9 10 11 12)(13 17 15 19)(14 18 16 20)
(1 12 3 10)(2 9 4 11)(5 6 7 8)(13 18 15 20)(14 19 16 17)(21 24 23 22)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 18 10)(6 19 11)(7 20 12)(8 17 9)
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,11,3,9)(2,10,4,12)(5,23,7,21)(6,22,8,24)(13,18,15,20)(14,17,16,19), (1,4,3,2)(5,23,7,21)(6,24,8,22)(9,10,11,12)(13,17,15,19)(14,18,16,20), (1,12,3,10)(2,9,4,11)(5,6,7,8)(13,18,15,20)(14,19,16,17)(21,24,23,22), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,18,10)(6,19,11)(7,20,12)(8,17,9)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,11,3,9)(2,10,4,12)(5,23,7,21)(6,22,8,24)(13,18,15,20)(14,17,16,19), (1,4,3,2)(5,23,7,21)(6,24,8,22)(9,10,11,12)(13,17,15,19)(14,18,16,20), (1,12,3,10)(2,9,4,11)(5,6,7,8)(13,18,15,20)(14,19,16,17)(21,24,23,22), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,18,10)(6,19,11)(7,20,12)(8,17,9) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,11,3,9),(2,10,4,12),(5,23,7,21),(6,22,8,24),(13,18,15,20),(14,17,16,19)], [(1,4,3,2),(5,23,7,21),(6,24,8,22),(9,10,11,12),(13,17,15,19),(14,18,16,20)], [(1,12,3,10),(2,9,4,11),(5,6,7,8),(13,18,15,20),(14,19,16,17),(21,24,23,22)], [(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,18,10),(6,19,11),(7,20,12),(8,17,9)]])
G:=TransitiveGroup(24,137);
Q8.A4 is a maximal subgroup of
Q8.4S4 Q8.5S4 Q16.A4 SD16.A4 Q8.6S4 Q8.7S4 2- 1+4⋊3C6 Ω4+ (𝔽3) Dic6.A4 Q8.A5 Dic10.A4
Q8.A4 is a maximal quotient of
C4○D4⋊C12 SL2(𝔽3)⋊6D4 Q8×SL2(𝔽3) 2+ 1+4⋊C9 Dic6.A4 Dic10.A4
Matrix representation of Q8.A4 ►in GL4(ℚ) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | -1 |
| -1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | 1 | 0 |
| 0 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| -1/2 | 1/2 | 1/2 | 1/2 |
| -1/2 | -1/2 | -1/2 | 1/2 |
| -1/2 | 1/2 | -1/2 | -1/2 |
| -1/2 | -1/2 | 1/2 | -1/2 |
G:=sub<GL(4,Rationals())| [0,0,-1,0,0,0,0,1,1,0,0,0,0,-1,0,0],[0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,1,0],[0,0,0,1,0,0,-1,0,0,1,0,0,-1,0,0,0],[0,0,1,0,0,0,0,1,-1,0,0,0,0,-1,0,0],[-1/2,-1/2,-1/2,-1/2,1/2,-1/2,1/2,-1/2,1/2,-1/2,-1/2,1/2,1/2,1/2,-1/2,-1/2] >;
Q8.A4 in GAP, Magma, Sage, TeX
Q_8.A_4
% in TeX
G:=Group("Q8.A4"); // GroupNames label
G:=SmallGroup(96,201);
// by ID
G=gap.SmallGroup(96,201);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-2,288,601,295,159,117,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=e^3=1,b^2=c^2=d^2=a^2,b*a*b^-1=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 Q8.A4 in TeX
Character table of Q8.A4 in TeX