direct product, non-abelian, soluble
Aliases: C2×SL2(𝔽3), Q8⋊C6, C22.2A4, (C2×Q8)⋊C3, C2.2(C2×A4), SmallGroup(48,32)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C2×SL2(𝔽3) |
Generators and relations for C2×SL2(𝔽3)
G = < a,b,c,d | a2=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >
Character table of C2×SL2(𝔽3)
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ5 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 2 | 2 | -2 | -2 | -1 | -1 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ8 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | -1 | 1 | -1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | -2 | 2 | ζ6 | ζ65 | 0 | 0 | ζ3 | ζ32 | ζ3 | ζ65 | ζ32 | ζ6 | complex lifted from SL2(𝔽3) |
| ρ10 | 2 | 2 | -2 | -2 | ζ65 | ζ6 | 0 | 0 | ζ32 | ζ3 | ζ6 | ζ32 | ζ65 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ11 | 2 | 2 | -2 | -2 | ζ6 | ζ65 | 0 | 0 | ζ3 | ζ32 | ζ65 | ζ3 | ζ6 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ12 | 2 | -2 | -2 | 2 | ζ65 | ζ6 | 0 | 0 | ζ32 | ζ3 | ζ32 | ζ6 | ζ3 | ζ65 | complex lifted from SL2(𝔽3) |
| ρ13 | 3 | -3 | 3 | -3 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
►On 16 points - transitive group 16T59
Generators in S16
(1 6)(2 7)(3 8)(4 5)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 12 3 10)(2 11 4 9)(5 13 7 15)(6 16 8 14)
(2 11 12)(4 9 10)(5 13 14)(7 15 16)
G:=sub<Sym(16)| (1,6)(2,7)(3,8)(4,5)(9,13)(10,14)(11,15)(12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,3,10)(2,11,4,9)(5,13,7,15)(6,16,8,14), (2,11,12)(4,9,10)(5,13,14)(7,15,16)>;
G:=Group( (1,6)(2,7)(3,8)(4,5)(9,13)(10,14)(11,15)(12,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,12,3,10)(2,11,4,9)(5,13,7,15)(6,16,8,14), (2,11,12)(4,9,10)(5,13,14)(7,15,16) );
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,5),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,12,3,10),(2,11,4,9),(5,13,7,15),(6,16,8,14)], [(2,11,12),(4,9,10),(5,13,14),(7,15,16)]])
G:=TransitiveGroup(16,59);
C2×SL2(𝔽3) is a maximal subgroup of
Q8⋊Dic3 Q8.D6 D4.A4 C24.7A4 Q8⋊SL2(𝔽3) C24⋊5A4 Q8⋊F7
C2×SL2(𝔽3) is a maximal quotient of C24.A4 C24.2A4 C24.3A4 Q8⋊F7
Matrix representation of C2×SL2(𝔽3) ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 1 | 0 | 0 |
| 0 | 1 | 5 |
| 0 | 10 | 12 |
| 1 | 0 | 0 |
| 0 | 12 | 2 |
| 0 | 12 | 1 |
| 3 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 10 | 3 |
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[1,0,0,0,1,10,0,5,12],[1,0,0,0,12,12,0,2,1],[3,0,0,0,1,10,0,0,3] >;
C2×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_2\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C2xSL(2,3)"); // GroupNames label
G:=SmallGroup(48,32);
// by ID
G=gap.SmallGroup(48,32);
# by ID
G:=PCGroup([5,-2,-3,-2,2,-2,97,72,188,133,58]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=d^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations
Export
Subgroup lattice of C2×SL2(𝔽3) in TeX
Character table of C2×SL2(𝔽3) in TeX