direct product, non-abelian, soluble
Aliases: C4×SL2(𝔽3), Q8⋊C12, (C4×Q8)⋊C3, C2.2(C4×A4), (C2×C4).1A4, C2.(C4.A4), (C2×Q8).1C6, C22.6(C2×A4), C2.(C2×SL2(𝔽3)), (C2×SL2(𝔽3)).3C2, SmallGroup(96,69)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C2×Q8 — C2×SL2(𝔽3) — C4×SL2(𝔽3) |
| Q8 — C4×SL2(𝔽3) |
Generators and relations for C4×SL2(𝔽3)
G = < a,b,c,d | a4=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >
Character table of C4×SL2(𝔽3)
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | -1 | 1 | i | -i | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | -i | -i | i | i | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | -1 | 1 | -i | i | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | i | i | -i | -i | i | -i | linear of order 4 |
| ρ9 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | -i | -i | i | i | -1 | 1 | -i | i | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ3 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | linear of order 12 |
| ρ10 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | i | i | -i | -i | -1 | 1 | i | -i | ζ32 | ζ3 | ζ6 | ζ65 | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ3 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | linear of order 12 |
| ρ11 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | -i | -i | i | i | -1 | 1 | -i | i | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ32 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | linear of order 12 |
| ρ12 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | i | i | -i | -i | -1 | 1 | i | -i | ζ3 | ζ32 | ζ65 | ζ6 | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ32 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | linear of order 12 |
| ρ13 | 2 | -2 | 2 | -2 | -1 | -1 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ14 | 2 | -2 | 2 | -2 | -1 | -1 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ15 | 2 | -2 | 2 | -2 | ζ65 | ζ6 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ6 | ζ65 | ζ3 | ζ32 | ζ6 | ζ65 | ζ3 | ζ65 | ζ32 | ζ6 | ζ32 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ16 | 2 | -2 | 2 | -2 | ζ6 | ζ65 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ65 | ζ6 | ζ32 | ζ3 | ζ65 | ζ6 | ζ32 | ζ6 | ζ3 | ζ65 | ζ3 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ17 | 2 | -2 | 2 | -2 | ζ6 | ζ65 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ65 | ζ6 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ32 | ζ65 | ζ3 | ζ65 | ζ6 | complex lifted from SL2(𝔽3) |
| ρ18 | 2 | -2 | 2 | -2 | ζ65 | ζ6 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ6 | ζ65 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ3 | ζ6 | ζ32 | ζ6 | ζ65 | complex lifted from SL2(𝔽3) |
| ρ19 | 2 | 2 | -2 | -2 | -1 | -1 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | i | -i | -i | i | i | -i | -i | i | complex lifted from C4.A4 |
| ρ20 | 2 | 2 | -2 | -2 | -1 | -1 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | -i | i | i | -i | -i | i | i | -i | complex lifted from C4.A4 |
| ρ21 | 2 | 2 | -2 | -2 | ζ65 | ζ6 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | ζ43ζ32 | ζ4ζ32 | ζ4ζ32 | ζ43ζ3 | complex lifted from C4.A4 |
| ρ22 | 2 | 2 | -2 | -2 | ζ65 | ζ6 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | ζ4ζ32 | ζ43ζ32 | ζ43ζ32 | ζ4ζ3 | complex lifted from C4.A4 |
| ρ23 | 2 | 2 | -2 | -2 | ζ6 | ζ65 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | ζ43ζ3 | ζ4ζ3 | ζ4ζ3 | ζ43ζ32 | complex lifted from C4.A4 |
| ρ24 | 2 | 2 | -2 | -2 | ζ6 | ζ65 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | ζ4ζ3 | ζ43ζ3 | ζ43ζ3 | ζ4ζ32 | complex lifted from C4.A4 |
| ρ25 | 3 | 3 | 3 | 3 | 0 | 0 | -3 | -3 | -3 | -3 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ26 | 3 | 3 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ27 | 3 | -3 | -3 | 3 | 0 | 0 | 3i | 3i | -3i | -3i | 1 | -1 | -i | i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4×A4 |
| ρ28 | 3 | -3 | -3 | 3 | 0 | 0 | -3i | -3i | 3i | 3i | 1 | -1 | i | -i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4×A4 |
►On 32 points
Generators in S32
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 21 20 28)(2 22 17 25)(3 23 18 26)(4 24 19 27)(5 16 30 9)(6 13 31 10)(7 14 32 11)(8 15 29 12)
(1 15 20 12)(2 16 17 9)(3 13 18 10)(4 14 19 11)(5 25 30 22)(6 26 31 23)(7 27 32 24)(8 28 29 21)
(5 16 22)(6 13 23)(7 14 24)(8 15 21)(9 25 30)(10 26 31)(11 27 32)(12 28 29)
G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,21,20,28)(2,22,17,25)(3,23,18,26)(4,24,19,27)(5,16,30,9)(6,13,31,10)(7,14,32,11)(8,15,29,12), (1,15,20,12)(2,16,17,9)(3,13,18,10)(4,14,19,11)(5,25,30,22)(6,26,31,23)(7,27,32,24)(8,28,29,21), (5,16,22)(6,13,23)(7,14,24)(8,15,21)(9,25,30)(10,26,31)(11,27,32)(12,28,29)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,21,20,28)(2,22,17,25)(3,23,18,26)(4,24,19,27)(5,16,30,9)(6,13,31,10)(7,14,32,11)(8,15,29,12), (1,15,20,12)(2,16,17,9)(3,13,18,10)(4,14,19,11)(5,25,30,22)(6,26,31,23)(7,27,32,24)(8,28,29,21), (5,16,22)(6,13,23)(7,14,24)(8,15,21)(9,25,30)(10,26,31)(11,27,32)(12,28,29) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,21,20,28),(2,22,17,25),(3,23,18,26),(4,24,19,27),(5,16,30,9),(6,13,31,10),(7,14,32,11),(8,15,29,12)], [(1,15,20,12),(2,16,17,9),(3,13,18,10),(4,14,19,11),(5,25,30,22),(6,26,31,23),(7,27,32,24),(8,28,29,21)], [(5,16,22),(6,13,23),(7,14,24),(8,15,21),(9,25,30),(10,26,31),(11,27,32),(12,28,29)]])
C4×SL2(𝔽3) is a maximal subgroup of
C2.U2(𝔽3) Q8⋊Dic6 CSU2(𝔽3)⋊C4 Q8.Dic6 Q8.D12 SL2(𝔽3)⋊Q8 Q8⋊D12 GL2(𝔽3)⋊C4 Q8.2D12 (C2×Q8)⋊C12 C4○D4⋊C12 SL2(𝔽3)⋊5D4 SL2(𝔽3)⋊6D4 SL2(𝔽3)⋊3Q8
Matrix representation of C4×SL2(𝔽3) ►in GL3(𝔽13) generated by
| 5 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 10 | 4 |
| 0 | 4 | 3 |
| 1 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 1 | 0 |
| 3 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 4 | 3 |
G:=sub<GL(3,GF(13))| [5,0,0,0,8,0,0,0,8],[1,0,0,0,10,4,0,4,3],[1,0,0,0,0,1,0,12,0],[3,0,0,0,1,4,0,0,3] >;
C4×SL2(𝔽3) in GAP, Magma, Sage, TeX
C_4\times {\rm SL}_2({\mathbb F}_3) % in TeX
G:=Group("C4xSL(2,3)"); // GroupNames label
G:=SmallGroup(96,69);
// by ID
G=gap.SmallGroup(96,69);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,2,-2,36,297,117,550,202,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=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 C4×SL2(𝔽3) in TeX
Character table of C4×SL2(𝔽3) in TeX