direct product, non-abelian, soluble
Aliases: S3×SL2(𝔽3), D6.A4, (S3×Q8)⋊C3, (C3×Q8)⋊C6, C2.3(S3×A4), C6.2(C2×A4), Q8⋊2(C3×S3), C3⋊(C2×SL2(𝔽3)), (C3×SL2(𝔽3))⋊3C2, SL2(ℤ/6ℤ), SmallGroup(144,128)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — C6 — C3×Q8 — C3×SL2(𝔽3) — S3×SL2(𝔽3) |
| C3×Q8 — S3×SL2(𝔽3) |
Generators and relations for S3×SL2(𝔽3)
G = < a,b,c,d,e | a3=b2=c4=e3=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >
Character table of S3×SL2(𝔽3)
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12 | |
| size | 1 | 1 | 3 | 3 | 2 | 4 | 4 | 8 | 8 | 6 | 18 | 2 | 4 | 4 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 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 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | 1 | linear of order 6 |
| ρ7 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 2 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | -2 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 0 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 0 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ10 | 2 | -2 | 2 | -2 | 2 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | 0 | -2 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ6 | ζ3 | ζ32 | 0 | complex lifted from SL2(𝔽3) |
| ρ11 | 2 | 2 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 2 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | -1 | complex lifted from C3×S3 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | 0 | -2 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ65 | ζ32 | ζ3 | 0 | complex lifted from SL2(𝔽3) |
| ρ13 | 2 | 2 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 2 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | -1 | complex lifted from C3×S3 |
| ρ14 | 2 | -2 | -2 | 2 | 2 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | 0 | -2 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | 0 | complex lifted from SL2(𝔽3) |
| ρ15 | 2 | -2 | -2 | 2 | 2 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | 0 | -2 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | 0 | complex lifted from SL2(𝔽3) |
| ρ16 | 3 | 3 | -3 | -3 | 3 | 0 | 0 | 0 | 0 | -1 | 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×A4 |
| ρ17 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from A4 |
| ρ18 | 4 | -4 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ19 | 4 | -4 | 0 | 0 | -2 | 1+√-3 | 1-√-3 | ζ3 | ζ32 | 0 | 0 | 2 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | -2 | 1-√-3 | 1+√-3 | ζ32 | ζ3 | 0 | 0 | 2 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal lifted from S3×A4 |
►On 24 points - transitive group 24T249
Generators in S24
(1 9 16)(2 10 13)(3 11 14)(4 12 15)(5 21 19)(6 22 20)(7 23 17)(8 24 18)
(1 3)(2 4)(5 17)(6 18)(7 19)(8 20)(9 14)(10 15)(11 16)(12 13)(21 23)(22 24)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 24 3 22)(2 23 4 21)(5 13 7 15)(6 16 8 14)(9 18 11 20)(10 17 12 19)
(2 23 24)(4 21 22)(5 6 15)(7 8 13)(10 17 18)(12 19 20)
G:=sub<Sym(24)| (1,9,16)(2,10,13)(3,11,14)(4,12,15)(5,21,19)(6,22,20)(7,23,17)(8,24,18), (1,3)(2,4)(5,17)(6,18)(7,19)(8,20)(9,14)(10,15)(11,16)(12,13)(21,23)(22,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,13,7,15)(6,16,8,14)(9,18,11,20)(10,17,12,19), (2,23,24)(4,21,22)(5,6,15)(7,8,13)(10,17,18)(12,19,20)>;
G:=Group( (1,9,16)(2,10,13)(3,11,14)(4,12,15)(5,21,19)(6,22,20)(7,23,17)(8,24,18), (1,3)(2,4)(5,17)(6,18)(7,19)(8,20)(9,14)(10,15)(11,16)(12,13)(21,23)(22,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,13,7,15)(6,16,8,14)(9,18,11,20)(10,17,12,19), (2,23,24)(4,21,22)(5,6,15)(7,8,13)(10,17,18)(12,19,20) );
G=PermutationGroup([[(1,9,16),(2,10,13),(3,11,14),(4,12,15),(5,21,19),(6,22,20),(7,23,17),(8,24,18)], [(1,3),(2,4),(5,17),(6,18),(7,19),(8,20),(9,14),(10,15),(11,16),(12,13),(21,23),(22,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,13,7,15),(6,16,8,14),(9,18,11,20),(10,17,12,19)], [(2,23,24),(4,21,22),(5,6,15),(7,8,13),(10,17,18),(12,19,20)]])
G:=TransitiveGroup(24,249);
S3×SL2(𝔽3) is a maximal subgroup of
D6.S4 D6.2S4 SL2(𝔽3).11D6 D12.A4 D18.A4 Q8⋊He3⋊C2
S3×SL2(𝔽3) is a maximal quotient of D18.A4 Q8⋊He3⋊C2
Matrix representation of S3×SL2(𝔽3) ►in GL4(𝔽5) generated by
| 3 | 2 | 2 | 1 |
| 4 | 4 | 1 | 1 |
| 1 | 3 | 3 | 3 |
| 2 | 3 | 4 | 3 |
| 1 | 1 | 4 | 2 |
| 4 | 0 | 2 | 2 |
| 0 | 1 | 0 | 2 |
| 3 | 0 | 2 | 4 |
| 0 | 4 | 1 | 2 |
| 0 | 3 | 1 | 1 |
| 1 | 2 | 3 | 3 |
| 4 | 3 | 4 | 4 |
| 1 | 3 | 1 | 4 |
| 1 | 2 | 0 | 4 |
| 3 | 4 | 0 | 1 |
| 3 | 3 | 1 | 2 |
| 3 | 3 | 0 | 0 |
| 4 | 1 | 0 | 0 |
| 1 | 1 | 0 | 1 |
| 2 | 0 | 4 | 4 |
G:=sub<GL(4,GF(5))| [3,4,1,2,2,4,3,3,2,1,3,4,1,1,3,3],[1,4,0,3,1,0,1,0,4,2,0,2,2,2,2,4],[0,0,1,4,4,3,2,3,1,1,3,4,2,1,3,4],[1,1,3,3,3,2,4,3,1,0,0,1,4,4,1,2],[3,4,1,2,3,1,1,0,0,0,0,4,0,0,1,4] >;
S3×SL2(𝔽3) in GAP, Magma, Sage, TeX
S_3\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("S3xSL(2,3)"); // GroupNames label
G:=SmallGroup(144,128);
// by ID
G=gap.SmallGroup(144,128);
# by ID
G:=PCGroup([6,-2,-3,-2,2,-3,-2,170,230,81,351,165,1444]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=e^3=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,e*c*e^-1=d,e*d*e^-1=c*d>;
// generators/relations
Export
Subgroup lattice of S3×SL2(𝔽3) in TeX
Character table of S3×SL2(𝔽3) in TeX