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G = Q8.D6order 96 = 25·3

2nd non-split extension by Q8 of D6 acting via D6/C2=S3

non-abelian, soluble

Aliases: Q8.2D6, C22.2S4, GL2(𝔽3)⋊1C2, CSU2(𝔽3)⋊1C2, SL2(𝔽3).2C22, C2.7(C2×S4), (C2×Q8)⋊2S3, (C2×SL2(𝔽3))⋊3C2, SmallGroup(96,190)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — Q8.D6
C1C2Q8SL2(𝔽3)GL2(𝔽3) — Q8.D6
SL2(𝔽3) — Q8.D6
C1C2C22

Generators and relations for Q8.D6
 G = < a,b,c,d | a4=c6=1, b2=d2=a2, bab-1=dad-1=a-1, cac-1=ab, cbc-1=a, dbd-1=a-1b, dcd-1=a2c-1 >

2C2
12C2
4C3
3C4
3C4
6C22
6C4
4C6
8C6
8S3
3D4
3Q8
3C8
3C2×C4
3Q8
3C8
6C2×C4
6D4
4D6
4C2×C6
4Dic3
3Q16
3SD16
3M4(2)
3C4○D4
3Q16
3SD16
4C3⋊D4
3C8.C22

Character table of Q8.D6

 class 12A2B2C34A4B4C6A6B6C8A8B
 size 11212866128881212
ρ11111111111111    trivial
ρ211-1-11-111-11-11-1    linear of order 2
ρ311-111-11-1-11-1-11    linear of order 2
ρ4111-1111-1111-1-1    linear of order 2
ρ522-20-1-2201-1100    orthogonal lifted from D6
ρ62220-1220-1-1-100    orthogonal lifted from S3
ρ7333-10-1-1-100011    orthogonal lifted from S4
ρ833-3101-1-10001-1    orthogonal lifted from C2×S4
ρ933-3-101-11000-11    orthogonal lifted from C2×S4
ρ1033310-1-11000-1-1    orthogonal lifted from S4
ρ114-400-200002000    symplectic faithful, Schur index 2
ρ124-4001000--3-1-300    complex faithful
ρ134-4001000-3-1--300    complex faithful

Permutation representations of Q8.D6
On 16 points - transitive group 16T187
Generators in S16
(1 10 4 15)(2 7 3 12)(5 14 16 9)(6 8 11 13)
(1 8 4 13)(2 5 3 16)(6 15 11 10)(7 9 12 14)
(1 2)(3 4)(5 6 7 8 9 10)(11 12 13 14 15 16)
(1 2 4 3)(5 6 16 11)(7 10 12 15)(8 14 13 9)

G:=sub<Sym(16)| (1,10,4,15)(2,7,3,12)(5,14,16,9)(6,8,11,13), (1,8,4,13)(2,5,3,16)(6,15,11,10)(7,9,12,14), (1,2)(3,4)(5,6,7,8,9,10)(11,12,13,14,15,16), (1,2,4,3)(5,6,16,11)(7,10,12,15)(8,14,13,9)>;

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

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

G:=TransitiveGroup(16,187);

On 16 points - transitive group 16T192
Generators in S16
(1 10 2 5)(3 16 4 13)(6 9 8 7)(11 12 14 15)
(1 8 2 6)(3 14 4 11)(5 7 10 9)(12 13 15 16)
(3 4)(5 6 7)(8 9 10)(11 12 13 14 15 16)
(1 3 2 4)(5 16 10 13)(6 12 8 15)(7 14 9 11)

G:=sub<Sym(16)| (1,10,2,5)(3,16,4,13)(6,9,8,7)(11,12,14,15), (1,8,2,6)(3,14,4,11)(5,7,10,9)(12,13,15,16), (3,4)(5,6,7)(8,9,10)(11,12,13,14,15,16), (1,3,2,4)(5,16,10,13)(6,12,8,15)(7,14,9,11)>;

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

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

G:=TransitiveGroup(16,192);

Q8.D6 is a maximal subgroup of
GL2(𝔽3)⋊C22  D4.4S4  D4.5S4  D6.S4  D6.2S4  SL2(𝔽3).D6  C22.S5  D10.1S4  D10.2S4  Q8.D30
Q8.D6 is a maximal quotient of
CSU2(𝔽3)⋊C4  Q8.Dic6  GL2(𝔽3)⋊C4  Q8.2D12  C23.14S4  C23.15S4  C23.16S4  Q8.D18  D6.S4  D6.2S4  SL2(𝔽3).D6  D10.1S4  D10.2S4  Q8.D30

Matrix representation of Q8.D6 in GL4(𝔽3) generated by

2002
0020
0100
2001
,
1002
0220
0210
2002
,
1001
0020
0110
0001
,
0020
1001
1000
0210
G:=sub<GL(4,GF(3))| [2,0,0,2,0,0,1,0,0,2,0,0,2,0,0,1],[1,0,0,2,0,2,2,0,0,2,1,0,2,0,0,2],[1,0,0,0,0,0,1,0,0,2,1,0,1,0,0,1],[0,1,1,0,0,0,0,2,2,0,0,1,0,1,0,0] >;

Q8.D6 in GAP, Magma, Sage, TeX

Q_8.D_6
% in TeX

G:=Group("Q8.D6");
// GroupNames label

G:=SmallGroup(96,190);
// by ID

G=gap.SmallGroup(96,190);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,2,-2,601,146,579,447,117,364,286,202,88]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^6=1,b^2=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,c*a*c^-1=a*b,c*b*c^-1=a,d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
// generators/relations

Export

Subgroup lattice of Q8.D6 in TeX
Character table of Q8.D6 in TeX

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