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
| SL2(𝔽3) — C2×GL2(𝔽3) |
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) |
►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
Export
Subgroup lattice of C2×GL2(𝔽3) in TeX
Character table of C2×GL2(𝔽3) in TeX