direct product, non-abelian, soluble
Aliases: C3×GL2(𝔽3), C6.9S4, SL2(𝔽3)⋊C6, Q8⋊(C3×S3), C2.3(C3×S4), (C3×Q8)⋊2S3, (C3×SL2(𝔽3))⋊4C2, SmallGroup(144,122)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C3×GL2(𝔽3) |
| SL2(𝔽3) — C3×GL2(𝔽3) |
Generators and relations for C3×GL2(𝔽3)
G = < a,b,c,d,e | a3=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece=b-1, dbd-1=bc, ebe=b2c, dcd-1=b, ede=d-1 >
Character table of C3×GL2(𝔽3)
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 12A | 12B | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 12 | 1 | 1 | 8 | 8 | 8 | 6 | 1 | 1 | 8 | 8 | 8 | 12 | 12 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ6 | ζ65 | -1 | -1 | ζ32 | ζ3 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ4 | 1 | 1 | -1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ65 | ζ6 | -1 | -1 | ζ3 | ζ32 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 2 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 0 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ9 | 2 | 2 | 0 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ10 | 2 | -2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | -√-2 | √-2 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | complex lifted from GL2(𝔽3) |
| ρ11 | 2 | -2 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | √-2 | -√-2 | 0 | 0 | √-2 | -√-2 | -√-2 | √-2 | complex lifted from GL2(𝔽3) |
| ρ12 | 2 | -2 | 0 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 0 | 1+√-3 | 1-√-3 | ζ32 | 1 | ζ3 | 0 | 0 | -√-2 | √-2 | 0 | 0 | ζ87ζ32+ζ85ζ32 | ζ83ζ32+ζ8ζ32 | ζ83ζ3+ζ8ζ3 | ζ87ζ3+ζ85ζ3 | complex faithful |
| ρ13 | 2 | -2 | 0 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 0 | 1-√-3 | 1+√-3 | ζ3 | 1 | ζ32 | 0 | 0 | -√-2 | √-2 | 0 | 0 | ζ87ζ3+ζ85ζ3 | ζ83ζ3+ζ8ζ3 | ζ83ζ32+ζ8ζ32 | ζ87ζ32+ζ85ζ32 | complex faithful |
| ρ14 | 2 | -2 | 0 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 0 | 1-√-3 | 1+√-3 | ζ3 | 1 | ζ32 | 0 | 0 | √-2 | -√-2 | 0 | 0 | ζ83ζ3+ζ8ζ3 | ζ87ζ3+ζ85ζ3 | ζ87ζ32+ζ85ζ32 | ζ83ζ32+ζ8ζ32 | complex faithful |
| ρ15 | 2 | -2 | 0 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 0 | 1+√-3 | 1-√-3 | ζ32 | 1 | ζ3 | 0 | 0 | √-2 | -√-2 | 0 | 0 | ζ83ζ32+ζ8ζ32 | ζ87ζ32+ζ85ζ32 | ζ87ζ3+ζ85ζ3 | ζ83ζ3+ζ8ζ3 | complex faithful |
| ρ16 | 3 | 3 | -1 | 3 | 3 | 0 | 0 | 0 | -1 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | 1 | 3 | 3 | 0 | 0 | 0 | -1 | 3 | 3 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ18 | 3 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | ζ6 | ζ65 | 1 | 1 | ζ6 | ζ65 | ζ3 | ζ3 | ζ32 | ζ32 | complex lifted from C3×S4 |
| ρ19 | 3 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | ζ3 | ζ32 | -1 | -1 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3×S4 |
| ρ20 | 3 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | ζ65 | ζ6 | 1 | 1 | ζ65 | ζ6 | ζ32 | ζ32 | ζ3 | ζ3 | complex lifted from C3×S4 |
| ρ21 | 3 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | ζ32 | ζ3 | -1 | -1 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3×S4 |
| ρ22 | 4 | -4 | 0 | 4 | 4 | 1 | 1 | 1 | 0 | -4 | -4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from GL2(𝔽3) |
| ρ23 | 4 | -4 | 0 | -2+2√-3 | -2-2√-3 | ζ3 | 1 | ζ32 | 0 | 2-2√-3 | 2+2√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | -4 | 0 | -2-2√-3 | -2+2√-3 | ζ32 | 1 | ζ3 | 0 | 2+2√-3 | 2-2√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T253
Generators in S24
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 19 11)(6 20 12)(7 17 9)(8 18 10)
(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 11 3 9)(2 10 4 12)(5 22 7 24)(6 21 8 23)(13 18 15 20)(14 17 16 19)
(2 11 10)(4 9 12)(5 8 21)(6 23 7)(13 19 18)(15 17 20)
(1 3)(2 11)(4 9)(5 21)(7 23)(13 19)(14 16)(15 17)(22 24)
G:=sub<Sym(24)| (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,19,11)(6,20,12)(7,17,9)(8,18,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,11,3,9)(2,10,4,12)(5,22,7,24)(6,21,8,23)(13,18,15,20)(14,17,16,19), (2,11,10)(4,9,12)(5,8,21)(6,23,7)(13,19,18)(15,17,20), (1,3)(2,11)(4,9)(5,21)(7,23)(13,19)(14,16)(15,17)(22,24)>;
G:=Group( (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,19,11)(6,20,12)(7,17,9)(8,18,10), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,11,3,9)(2,10,4,12)(5,22,7,24)(6,21,8,23)(13,18,15,20)(14,17,16,19), (2,11,10)(4,9,12)(5,8,21)(6,23,7)(13,19,18)(15,17,20), (1,3)(2,11)(4,9)(5,21)(7,23)(13,19)(14,16)(15,17)(22,24) );
G=PermutationGroup([[(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,19,11),(6,20,12),(7,17,9),(8,18,10)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,11,3,9),(2,10,4,12),(5,22,7,24),(6,21,8,23),(13,18,15,20),(14,17,16,19)], [(2,11,10),(4,9,12),(5,8,21),(6,23,7),(13,19,18),(15,17,20)], [(1,3),(2,11),(4,9),(5,21),(7,23),(13,19),(14,16),(15,17),(22,24)]])
G:=TransitiveGroup(24,253);
C3×GL2(𝔽3) is a maximal subgroup of
Dic3.4S4 GL2(𝔽3)⋊S3 D6.S4 C32⋊3GL2(𝔽3)
C3×GL2(𝔽3) is a maximal quotient of C32.GL2(𝔽3) C32⋊2GL2(𝔽3)
Matrix representation of C3×GL2(𝔽3) ►in GL2(𝔽19) generated by
| 11 | 0 |
| 0 | 11 |
| 6 | 1 |
| 1 | 13 |
| 0 | 18 |
| 1 | 0 |
| 2 | 3 |
| 4 | 16 |
| 1 | 13 |
| 0 | 18 |
G:=sub<GL(2,GF(19))| [11,0,0,11],[6,1,1,13],[0,1,18,0],[2,4,3,16],[1,0,13,18] >;
C3×GL2(𝔽3) in GAP, Magma, Sage, TeX
C_3\times {\rm GL}_2({\mathbb F}_3) % in TeX
G:=Group("C3xGL(2,3)"); // GroupNames label
G:=SmallGroup(144,122);
// by ID
G=gap.SmallGroup(144,122);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,2,-2,218,867,447,117,544,286,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e=b^-1,d*b*d^-1=b*c,e*b*e=b^2*c,d*c*d^-1=b,e*d*e=d^-1>;
// generators/relations
Export
Subgroup lattice of C3×GL2(𝔽3) in TeX
Character table of C3×GL2(𝔽3) in TeX