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G = C2×GL2(𝔽3)  order 96 = 25·3

Direct product of C2 and GL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C2×GL2(𝔽3), Q8⋊D6, C22.5S4, SL2(𝔽3)⋊1C22, C2.6(C2×S4), (C2×Q8)⋊1S3, (C2×SL2(𝔽3))⋊2C2, SmallGroup(96,189)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — C2×GL2(𝔽3)
C1C2Q8SL2(𝔽3)GL2(𝔽3) — C2×GL2(𝔽3)
SL2(𝔽3) — C2×GL2(𝔽3)
C1C22

Generators and relations for C2×GL2(𝔽3)
 G = < a,b,c,d,e | a2=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece=b-1, dbd-1=bc, ebe=b2c, dcd-1=b, ede=d-1 >

12C2
12C2
4C3
3C4
3C4
6C22
6C22
12C22
12C22
4S3
4S3
4S3
4S3
4C6
4C6
4C6
3C8
3C2×C4
3C8
3D4
3Q8
3D4
6C23
6D4
4D6
4D6
4D6
4D6
4C2×C6
4D6
4D6
3SD16
3SD16
3SD16
3C2×C8
3C2×D4
3SD16
4C22×S3
3C2×SD16

Character table of C2×GL2(𝔽3)

 class 12A2B2C2D2E34A4B6A6B6C8A8B8C8D
 size 111112128668886666
ρ11111111111111111    trivial
ρ211-1-1-1111-1-11-111-1-1    linear of order 2
ρ31111-1-1111111-1-1-1-1    linear of order 2
ρ411-1-11-111-1-11-1-1-111    linear of order 2
ρ5222200-122-1-1-10000    orthogonal lifted from S3
ρ622-2-200-12-21-110000    orthogonal lifted from D6
ρ72-22-200-100-111-2--2--2-2    complex lifted from GL2(𝔽3)
ρ82-2-2200-10011-1-2--2-2--2    complex lifted from GL2(𝔽3)
ρ92-2-2200-10011-1--2-2--2-2    complex lifted from GL2(𝔽3)
ρ102-22-200-100-111--2-2-2--2    complex lifted from GL2(𝔽3)
ρ113333-1-10-1-10001111    orthogonal lifted from S4
ρ1233-3-3-110-11000-1-111    orthogonal lifted from C2×S4
ρ133333110-1-1000-1-1-1-1    orthogonal lifted from S4
ρ1433-3-31-10-1100011-1-1    orthogonal lifted from C2×S4
ρ154-4-4400100-1-110000    orthogonal lifted from GL2(𝔽3)
ρ164-44-4001001-1-10000    orthogonal lifted from GL2(𝔽3)

Permutation representations of C2×GL2(𝔽3)
On 16 points - transitive group 16T188
Generators in S16
(1 16)(2 13)(3 14)(4 15)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 11 3 9)(2 10 4 12)(5 14 7 16)(6 13 8 15)
(2 11 10)(4 9 12)(5 8 13)(6 15 7)
(1 16)(2 7)(3 14)(4 5)(6 10)(8 12)(9 13)(11 15)

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

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

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

G:=TransitiveGroup(16,188);

C2×GL2(𝔽3) is a maximal subgroup of   Q8⋊D12  GL2(𝔽3)⋊C4  Q8.2D12  C23.16S4  SL2(𝔽3)⋊D4  D4.4S4
C2×GL2(𝔽3) is a maximal quotient of   Q8⋊Dic6  Q8⋊D12  C23.16S4

Matrix representation of C2×GL2(𝔽3) in GL3(𝔽73) generated by

7200
0720
0072
,
100
04544
01728
,
100
04556
02928
,
100
0072
0172
,
100
001
010
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[1,0,0,0,45,17,0,44,28],[1,0,0,0,45,29,0,56,28],[1,0,0,0,0,1,0,72,72],[1,0,0,0,0,1,0,1,0] >;

C2×GL2(𝔽3) in GAP, Magma, Sage, TeX

C_2\times {\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("C2xGL(2,3)");
// GroupNames label

G:=SmallGroup(96,189);
// by ID

G=gap.SmallGroup(96,189);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e=b^-1,d*b*d^-1=b*c,e*b*e=b^2*c,d*c*d^-1=b,e*d*e=d^-1>;
// generators/relations

Export

Subgroup lattice of C2×GL2(𝔽3) in TeX
Character table of C2×GL2(𝔽3) in TeX

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