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G = Q8.A4order 96 = 25·3

The non-split extension by Q8 of A4 acting through Inn(Q8)

non-abelian, soluble

Aliases: Q8.A4, Q8SL2(𝔽3), 2+ 1+41C3, SL2(𝔽3)⋊4C22, C4○D4.C6, C4.A43C2, C4.2(C2×A4), Q8.2(C2×C6), C2.6(C22×A4), SmallGroup(96,201)

Series: Derived Chief Lower central Upper central

C1C2Q8 — Q8.A4
C1C2Q8SL2(𝔽3)C4.A4 — Q8.A4
Q8 — Q8.A4
C1C2Q8

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 >

6C2
6C2
6C2
4C3
3C22
3C22
3C22
3C4
12C22
12C22
4C6
3D4
3C2×C4
3C23
3C2×C4
3D4
3C2×C4
3D4
3D4
3D4
3D4
3C23
4C12
4C12
4C12
3C2×D4
3C2×D4
3C2×D4
3C4○D4
4C3×Q8

Character table of Q8.A4

 class 12A2B2C2D3A3B4A4B4C4D6A6B12A12B12C12D12E12F
 size 1166644222644888888
ρ11111111111111111111    trivial
ρ211-11-1111-1-1111-11-1-1-11    linear of order 2
ρ311-1-1111-11-1111-1-111-1-1    linear of order 2
ρ4111-1-111-1-111111-1-1-11-1    linear of order 2
ρ511111ζ32ζ31111ζ3ζ32ζ3ζ3ζ32ζ3ζ32ζ32    linear of order 3
ρ611-1-11ζ3ζ32-11-11ζ32ζ3ζ6ζ6ζ3ζ32ζ65ζ65    linear of order 6
ρ711111ζ3ζ321111ζ32ζ3ζ32ζ32ζ3ζ32ζ3ζ3    linear of order 3
ρ8111-1-1ζ32ζ3-1-111ζ3ζ32ζ3ζ65ζ6ζ65ζ32ζ6    linear of order 6
ρ911-11-1ζ32ζ31-1-11ζ3ζ32ζ65ζ3ζ6ζ65ζ6ζ32    linear of order 6
ρ10111-1-1ζ3ζ32-1-111ζ32ζ3ζ32ζ6ζ65ζ6ζ3ζ65    linear of order 6
ρ1111-11-1ζ3ζ321-1-11ζ32ζ3ζ6ζ32ζ65ζ6ζ65ζ3    linear of order 6
ρ1211-1-11ζ32ζ3-11-11ζ3ζ32ζ65ζ65ζ32ζ3ζ6ζ6    linear of order 6
ρ13331-11003-3-3-100000000    orthogonal lifted from C2×A4
ρ1433-11100-3-33-100000000    orthogonal lifted from C2×A4
ρ1533-1-1-100333-100000000    orthogonal lifted from A4
ρ163311-100-33-3-100000000    orthogonal lifted from C2×A4
ρ174-4000-2-2000022000000    orthogonal faithful
ρ184-40001--31+-30000-1--3-1+-3000000    complex faithful
ρ194-40001+-31--30000-1+-3-1--3000000    complex faithful

Permutation representations of Q8.A4
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+43C6  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

0010
000-1
-1000
0100
,
0100
-1000
0001
00-10
,
000-1
0010
0-100
1000
,
00-10
000-1
1000
0100
,
-1/21/21/21/2
-1/2-1/2-1/21/2
-1/21/2-1/2-1/2
-1/2-1/21/2-1/2
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

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