direct product, non-abelian, soluble
Aliases: C3×SL2(𝔽3), Q8⋊C32, C6.3A4, C2.(C3×A4), (C3×Q8)⋊C3, SmallGroup(72,25)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8 — C3×SL2(𝔽3) |
Generators and relations for C3×SL2(𝔽3)
G = < a,b,c,d | a3=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >
Character table of C3×SL2(𝔽3)
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 12A | 12B | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | |
| 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 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ7 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ8 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | 1 | 1 | linear of order 3 |
| ρ10 | 2 | -2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ11 | 2 | -2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | ζ6 | -1 | ζ65 | 0 | 1-√-3 | 1+√-3 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | 0 | 0 | complex faithful |
| ρ12 | 2 | -2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | ζ65 | -1 | ζ6 | 0 | 1+√-3 | 1-√-3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | 0 | 0 | complex faithful |
| ρ13 | 2 | -2 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | ζ6 | -1 | ζ65 | -1 | 0 | 1-√-3 | 1+√-3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | 0 | 0 | complex faithful |
| ρ14 | 2 | -2 | 2 | 2 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | -2 | -2 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | 0 | 0 | complex lifted from SL2(𝔽3) |
| ρ15 | 2 | -2 | -1+√-3 | -1-√-3 | -1 | -1 | ζ65 | ζ65 | ζ6 | ζ6 | 0 | 1-√-3 | 1+√-3 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 0 | 0 | complex faithful |
| ρ16 | 2 | -2 | 2 | 2 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | -2 | -2 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | 0 | 0 | complex lifted from SL2(𝔽3) |
| ρ17 | 2 | -2 | -1-√-3 | -1+√-3 | -1 | -1 | ζ6 | ζ6 | ζ65 | ζ65 | 0 | 1+√-3 | 1-√-3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 0 | 0 | complex faithful |
| ρ18 | 2 | -2 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | ζ65 | -1 | ζ6 | -1 | 0 | 1+√-3 | 1-√-3 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | 0 | 0 | complex faithful |
| ρ19 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from A4 |
| ρ20 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | complex lifted from C3×A4 |
| ρ21 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | complex lifted from C3×A4 |
►On 24 points - transitive group 24T70
Generators in S24
(1 8 18)(2 5 19)(3 6 20)(4 7 17)(9 23 15)(10 24 16)(11 21 13)(12 22 14)
(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 12 3 10)(2 11 4 9)(5 21 7 23)(6 24 8 22)(13 17 15 19)(14 20 16 18)
(1 8 18)(2 21 14)(3 6 20)(4 23 16)(5 13 12)(7 15 10)(9 24 17)(11 22 19)
G:=sub<Sym(24)| (1,8,18)(2,5,19)(3,6,20)(4,7,17)(9,23,15)(10,24,16)(11,21,13)(12,22,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,12,3,10)(2,11,4,9)(5,21,7,23)(6,24,8,22)(13,17,15,19)(14,20,16,18), (1,8,18)(2,21,14)(3,6,20)(4,23,16)(5,13,12)(7,15,10)(9,24,17)(11,22,19)>;
G:=Group( (1,8,18)(2,5,19)(3,6,20)(4,7,17)(9,23,15)(10,24,16)(11,21,13)(12,22,14), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,12,3,10)(2,11,4,9)(5,21,7,23)(6,24,8,22)(13,17,15,19)(14,20,16,18), (1,8,18)(2,21,14)(3,6,20)(4,23,16)(5,13,12)(7,15,10)(9,24,17)(11,22,19) );
G=PermutationGroup([[(1,8,18),(2,5,19),(3,6,20),(4,7,17),(9,23,15),(10,24,16),(11,21,13),(12,22,14)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,12,3,10),(2,11,4,9),(5,21,7,23),(6,24,8,22),(13,17,15,19),(14,20,16,18)], [(1,8,18),(2,21,14),(3,6,20),(4,23,16),(5,13,12),(7,15,10),(9,24,17),(11,22,19)]])
G:=TransitiveGroup(24,70);
C3×SL2(𝔽3) is a maximal subgroup of
C6.5S4 C6.6S4 Dic3.A4 C18.A4 Q8⋊He3 Ω4+ (𝔽3)
C3×SL2(𝔽3) is a maximal quotient of C18.A4 Q8⋊3- 1+2 Q8⋊He3
Matrix representation of C3×SL2(𝔽3) ►in GL2(𝔽7) generated by
| 4 | 0 |
| 0 | 4 |
| 3 | 2 |
| 2 | 4 |
| 0 | 1 |
| 6 | 0 |
| 2 | 0 |
| 4 | 1 |
G:=sub<GL(2,GF(7))| [4,0,0,4],[3,2,2,4],[0,6,1,0],[2,4,0,1] >;
C3×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_3\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C3xSL(2,3)"); // GroupNames label
G:=SmallGroup(72,25);
// by ID
G=gap.SmallGroup(72,25);
# by ID
G:=PCGroup([5,-3,-3,-2,2,-2,272,72,543,133,58]);
// Polycyclic
G:=Group<a,b,c,d|a^3=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 C3×SL2(𝔽3) in TeX
Character table of C3×SL2(𝔽3) in TeX