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## G = C2×GL2(𝔽3)  order 96 = 25·3

### Direct product of C2 and GL2(𝔽3)

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

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C2×GL2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3) — C2×GL2(𝔽3)
 Lower central SL2(𝔽3) — C2×GL2(𝔽3)
 Upper central C1 — C22

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
3C8
3D4
3Q8
3D4
6C23
6D4
4D6
4D6
4D6
4D6
4D6
4D6
3SD16
3SD16
3SD16
3SD16

Character table of C2×GL2(𝔽3)

 class 1 2A 2B 2C 2D 2E 3 4A 4B 6A 6B 6C 8A 8B 8C 8D size 1 1 1 1 12 12 8 6 6 8 8 8 6 6 6 6 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 2 2 0 0 -1 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ6 2 2 -2 -2 0 0 -1 2 -2 1 -1 1 0 0 0 0 orthogonal lifted from D6 ρ7 2 -2 2 -2 0 0 -1 0 0 -1 1 1 √-2 -√-2 -√-2 √-2 complex lifted from GL2(𝔽3) ρ8 2 -2 -2 2 0 0 -1 0 0 1 1 -1 √-2 -√-2 √-2 -√-2 complex lifted from GL2(𝔽3) ρ9 2 -2 -2 2 0 0 -1 0 0 1 1 -1 -√-2 √-2 -√-2 √-2 complex lifted from GL2(𝔽3) ρ10 2 -2 2 -2 0 0 -1 0 0 -1 1 1 -√-2 √-2 √-2 -√-2 complex lifted from GL2(𝔽3) ρ11 3 3 3 3 -1 -1 0 -1 -1 0 0 0 1 1 1 1 orthogonal lifted from S4 ρ12 3 3 -3 -3 -1 1 0 -1 1 0 0 0 -1 -1 1 1 orthogonal lifted from C2×S4 ρ13 3 3 3 3 1 1 0 -1 -1 0 0 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ14 3 3 -3 -3 1 -1 0 -1 1 0 0 0 1 1 -1 -1 orthogonal lifted from C2×S4 ρ15 4 -4 -4 4 0 0 1 0 0 -1 -1 1 0 0 0 0 orthogonal lifted from GL2(𝔽3) ρ16 4 -4 4 -4 0 0 1 0 0 1 -1 -1 0 0 0 0 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

 72 0 0 0 72 0 0 0 72
,
 1 0 0 0 45 44 0 17 28
,
 1 0 0 0 45 56 0 29 28
,
 1 0 0 0 0 72 0 1 72
,
 1 0 0 0 0 1 0 1 0
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

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