Aliases: C23.1SL2(𝔽3), C4.10C42⋊C3, C4.2(C42⋊C3), (C22×C4).6A4, C2.(C23.3A4), SmallGroup(192,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C4.10C42 — C23.SL2(𝔽3) |
Generators and relations for C23.SL2(𝔽3)
G = < a,b,c,d,e,f | a2=b2=c2=f3=1, d4=c, e2=bd2, ab=ba, eae-1=fbf-1=ac=ca, ad=da, faf-1=abc, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, cf=fc, ede-1=ad3, fdf-1=be, fef-1=acde >
Character table of C23.SL2(𝔽3)
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 6 | 16 | 16 | 1 | 1 | 6 | 16 | 16 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ4 | 2 | 2 | 2 | -1 | -1 | -2 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ5 | 2 | 2 | 2 | ζ6 | ζ65 | -2 | -2 | -2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ3 | ζ3 | ζ32 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ6 | 2 | 2 | 2 | ζ65 | ζ6 | -2 | -2 | -2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ32 | ζ32 | ζ3 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ7 | 3 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ8 | 3 | 3 | -1 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | -1-2i | 1 | -1+2i | 1 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ9 | 3 | 3 | -1 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 1 | -1-2i | 1 | -1+2i | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ10 | 3 | 3 | -1 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 1 | -1+2i | 1 | -1-2i | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ11 | 3 | 3 | -1 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | -1+2i | 1 | -1-2i | 1 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ12 | 4 | -4 | 0 | 1 | 1 | 4i | -4i | 0 | -1 | -1 | 0 | 0 | 0 | 0 | i | -i | i | -i | complex faithful |
| ρ13 | 4 | -4 | 0 | 1 | 1 | -4i | 4i | 0 | -1 | -1 | 0 | 0 | 0 | 0 | -i | i | -i | i | complex faithful |
| ρ14 | 4 | -4 | 0 | ζ3 | ζ32 | -4i | 4i | 0 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | complex faithful |
| ρ15 | 4 | -4 | 0 | ζ32 | ζ3 | -4i | 4i | 0 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | complex faithful |
| ρ16 | 4 | -4 | 0 | ζ3 | ζ32 | 4i | -4i | 0 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | complex faithful |
| ρ17 | 4 | -4 | 0 | ζ32 | ζ3 | 4i | -4i | 0 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | complex faithful |
| ρ18 | 6 | 6 | -2 | 0 | 0 | -6 | -6 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23.3A4 |
►On 16 points - transitive group 16T439
Generators in S16
(9 13)(10 14)(11 15)(12 16)
(1 5)(3 7)(10 14)(12 16)
(1 5)(2 6)(3 7)(4 8)(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 11 7 13 5 15 3 9)(2 14 4 12 6 10 8 16)
(2 14 15)(4 12 13)(6 10 11)(8 16 9)
G:=sub<Sym(16)| (9,13)(10,14)(11,15)(12,16), (1,5)(3,7)(10,14)(12,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11,7,13,5,15,3,9)(2,14,4,12,6,10,8,16), (2,14,15)(4,12,13)(6,10,11)(8,16,9)>;
G:=Group( (9,13)(10,14)(11,15)(12,16), (1,5)(3,7)(10,14)(12,16), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,11,7,13,5,15,3,9)(2,14,4,12,6,10,8,16), (2,14,15)(4,12,13)(6,10,11)(8,16,9) );
G=PermutationGroup([[(9,13),(10,14),(11,15),(12,16)], [(1,5),(3,7),(10,14),(12,16)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,11,7,13,5,15,3,9),(2,14,4,12,6,10,8,16)], [(2,14,15),(4,12,13),(6,10,11),(8,16,9)]])
G:=TransitiveGroup(16,439);
Matrix representation of C23.SL2(𝔽3) ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 4 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 4 | 0 | 0 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,2,0,0,4,0,0,0,0,0,0,4,0,0,3,0],[0,0,1,0,0,0,0,4,2,0,0,0,0,2,0,0],[1,0,0,0,0,0,0,4,0,3,0,0,0,0,3,0] >;
C23.SL2(𝔽3) in GAP, Magma, Sage, TeX
C_2^3.{\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C2^3.SL(2,3)"); // GroupNames label
G:=SmallGroup(192,4);
// by ID
G=gap.SmallGroup(192,4);
# by ID
G:=PCGroup([7,-3,-2,2,-2,2,-2,-2,85,176,695,387,58,4707,1018,248,2944,1411,718,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=f^3=1,d^4=c,e^2=b*d^2,a*b=b*a,e*a*e^-1=f*b*f^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e^-1=a*d^3,f*d*f^-1=b*e,f*e*f^-1=a*c*d*e>;
// generators/relations
Export
Subgroup lattice of C23.SL2(𝔽3) in TeX
Character table of C23.SL2(𝔽3) in TeX