direct product, non-abelian, soluble
Aliases: C2×ASL2(𝔽3), PSU3(𝔽2)⋊C6, C3⋊S3⋊SL2(𝔽3), (C3×C6)⋊SL2(𝔽3), (C2×PSU3(𝔽2))⋊C3, C32⋊(C2×SL2(𝔽3)), C3⋊S3.(C2×A4), (C2×C3⋊S3).A4, SmallGroup(432,735)
Series: Derived ►Chief ►Lower central ►Upper central
| PSU3(𝔽2) — C2×ASL2(𝔽3) |
Generators and relations for C2×ASL2(𝔽3)
G = < a,b,c,d,e,f | a2=b3=c3=d4=f3=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, cf=fc, ede-1=d-1, fdf-1=e, fef-1=de >
9C2
9C2
4C3
12C3
24C3
9C22
27C4
27C4
4C6
12S3
12C6
12S3
24C6
36C6
36C6
4C32
8C32
9Q8
27Q8
27C2×C4
12D6
36C2×C6
12C3×S3
12C3×S3
4He3
12S3×C6
Character table of C2×ASL2(𝔽3)
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | |
| size | 1 | 1 | 9 | 9 | 8 | 12 | 12 | 24 | 24 | 54 | 54 | 8 | 12 | 12 | 24 | 24 | 36 | 36 | 36 | 36 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | -1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | -1 | -1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ32 | ζ3 | ζ6 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | -1 | -1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ3 | ζ32 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ8 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | 2 | -2 | 2 | ζ6 | ζ65 | ζ6 | ζ65 | 0 | 0 | -2 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | complex lifted from SL2(𝔽3) |
| ρ10 | 2 | 2 | -2 | -2 | 2 | ζ65 | ζ6 | ζ65 | ζ6 | 0 | 0 | 2 | ζ6 | ζ65 | ζ6 | ζ65 | ζ32 | ζ3 | ζ32 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ11 | 2 | 2 | -2 | -2 | 2 | ζ6 | ζ65 | ζ6 | ζ65 | 0 | 0 | 2 | ζ65 | ζ6 | ζ65 | ζ6 | ζ3 | ζ32 | ζ3 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ12 | 2 | -2 | 2 | -2 | 2 | ζ65 | ζ6 | ζ65 | ζ6 | 0 | 0 | -2 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | complex lifted from SL2(𝔽3) |
| ρ13 | 3 | -3 | -3 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | 1 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ15 | 8 | 8 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from ASL2(𝔽3) |
| ρ16 | 8 | -8 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ17 | 8 | -8 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 1 | 1-√-3 | 1+√-3 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | complex faithful |
| ρ18 | 8 | 8 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | complex lifted from ASL2(𝔽3) |
| ρ19 | 8 | -8 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 1 | 1+√-3 | 1-√-3 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 8 | 8 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | complex lifted from ASL2(𝔽3) |
►On 18 points - transitive group 18T151
Generators in S18
(1 2)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)
(1 17 15)(2 9 7)(3 6 16)(4 5 18)(8 11 14)(10 12 13)
(1 18 16)(2 10 8)(3 17 4)(5 6 15)(7 13 14)(9 12 11)
(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(3 15 5 17)(4 18 6 16)(7 13 9 11)(8 12 10 14)
(3 4 17)(5 6 15)(7 13 14)(9 11 12)
G:=sub<Sym(18)| (1,2)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18), (1,17,15)(2,9,7)(3,6,16)(4,5,18)(8,11,14)(10,12,13), (1,18,16)(2,10,8)(3,17,4)(5,6,15)(7,13,14)(9,12,11), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (3,15,5,17)(4,18,6,16)(7,13,9,11)(8,12,10,14), (3,4,17)(5,6,15)(7,13,14)(9,11,12)>;
G:=Group( (1,2)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18), (1,17,15)(2,9,7)(3,6,16)(4,5,18)(8,11,14)(10,12,13), (1,18,16)(2,10,8)(3,17,4)(5,6,15)(7,13,14)(9,12,11), (3,4,5,6)(7,8,9,10)(11,12,13,14)(15,16,17,18), (3,15,5,17)(4,18,6,16)(7,13,9,11)(8,12,10,14), (3,4,17)(5,6,15)(7,13,14)(9,11,12) );
G=PermutationGroup([[(1,2),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18)], [(1,17,15),(2,9,7),(3,6,16),(4,5,18),(8,11,14),(10,12,13)], [(1,18,16),(2,10,8),(3,17,4),(5,6,15),(7,13,14),(9,12,11)], [(3,4,5,6),(7,8,9,10),(11,12,13,14),(15,16,17,18)], [(3,15,5,17),(4,18,6,16),(7,13,9,11),(8,12,10,14)], [(3,4,17),(5,6,15),(7,13,14),(9,11,12)]])
G:=TransitiveGroup(18,151);
►On 24 points - transitive group 24T1320
Generators in S24
(1 4)(2 3)(5 7)(6 8)(9 23)(10 24)(11 21)(12 22)(13 18)(14 19)(15 20)(16 17)
(1 9 11)(2 10 12)(3 24 22)(4 23 21)(6 13 15)(8 18 20)
(1 11 9)(2 10 12)(3 24 22)(4 21 23)(5 14 16)(7 19 17)
(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 8)(2 7)(3 5)(4 6)(9 18 11 20)(10 17 12 19)(13 21 15 23)(14 24 16 22)
(1 7 2)(3 4 5)(9 17 12)(10 11 19)(14 24 21)(16 22 23)
G:=sub<Sym(24)| (1,4)(2,3)(5,7)(6,8)(9,23)(10,24)(11,21)(12,22)(13,18)(14,19)(15,20)(16,17), (1,9,11)(2,10,12)(3,24,22)(4,23,21)(6,13,15)(8,18,20), (1,11,9)(2,10,12)(3,24,22)(4,21,23)(5,14,16)(7,19,17), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,5)(4,6)(9,18,11,20)(10,17,12,19)(13,21,15,23)(14,24,16,22), (1,7,2)(3,4,5)(9,17,12)(10,11,19)(14,24,21)(16,22,23)>;
G:=Group( (1,4)(2,3)(5,7)(6,8)(9,23)(10,24)(11,21)(12,22)(13,18)(14,19)(15,20)(16,17), (1,9,11)(2,10,12)(3,24,22)(4,23,21)(6,13,15)(8,18,20), (1,11,9)(2,10,12)(3,24,22)(4,21,23)(5,14,16)(7,19,17), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,5)(4,6)(9,18,11,20)(10,17,12,19)(13,21,15,23)(14,24,16,22), (1,7,2)(3,4,5)(9,17,12)(10,11,19)(14,24,21)(16,22,23) );
G=PermutationGroup([[(1,4),(2,3),(5,7),(6,8),(9,23),(10,24),(11,21),(12,22),(13,18),(14,19),(15,20),(16,17)], [(1,9,11),(2,10,12),(3,24,22),(4,23,21),(6,13,15),(8,18,20)], [(1,11,9),(2,10,12),(3,24,22),(4,21,23),(5,14,16),(7,19,17)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8),(2,7),(3,5),(4,6),(9,18,11,20),(10,17,12,19),(13,21,15,23),(14,24,16,22)], [(1,7,2),(3,4,5),(9,17,12),(10,11,19),(14,24,21),(16,22,23)]])
G:=TransitiveGroup(24,1320);
►On 24 points - transitive group 24T1321
Generators in S24
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 20)(14 17)(15 18)(16 19)(21 23)(22 24)
(1 19 14)(3 16 17)(5 10 21)(6 22 11)(7 23 12)(8 9 24)
(2 20 15)(4 13 18)(5 10 21)(6 11 22)(7 23 12)(8 24 9)
(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 22 3 24)(2 21 4 23)(5 18 7 20)(6 17 8 19)(9 14 11 16)(10 13 12 15)
(2 21 22)(4 23 24)(5 6 20)(7 8 18)(9 13 12)(10 11 15)
G:=sub<Sym(24)| (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24), (1,19,14)(3,16,17)(5,10,21)(6,22,11)(7,23,12)(8,9,24), (2,20,15)(4,13,18)(5,10,21)(6,11,22)(7,23,12)(8,24,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22,3,24)(2,21,4,23)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15), (2,21,22)(4,23,24)(5,6,20)(7,8,18)(9,13,12)(10,11,15)>;
G:=Group( (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24), (1,19,14)(3,16,17)(5,10,21)(6,22,11)(7,23,12)(8,9,24), (2,20,15)(4,13,18)(5,10,21)(6,11,22)(7,23,12)(8,24,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,22,3,24)(2,21,4,23)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15), (2,21,22)(4,23,24)(5,6,20)(7,8,18)(9,13,12)(10,11,15) );
G=PermutationGroup([[(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,20),(14,17),(15,18),(16,19),(21,23),(22,24)], [(1,19,14),(3,16,17),(5,10,21),(6,22,11),(7,23,12),(8,9,24)], [(2,20,15),(4,13,18),(5,10,21),(6,11,22),(7,23,12),(8,24,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,22,3,24),(2,21,4,23),(5,18,7,20),(6,17,8,19),(9,14,11,16),(10,13,12,15)], [(2,21,22),(4,23,24),(5,6,20),(7,8,18),(9,13,12),(10,11,15)]])
G:=TransitiveGroup(24,1321);
Matrix representation of C2×ASL2(𝔽3) ►in GL8(ℤ)
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,-1,0,0,0,1,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,1,0,0,0,0,0,1,-1,0,0,0,0,0,0],[0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,1,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,1,-1,0,0,0],[1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1],[1,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
C2×ASL2(𝔽3) in GAP, Magma, Sage, TeX
C_2\times {\rm ASL}_2({\mathbb F}_3) % in TeX
G:=Group("C2xASL(2,3)"); // GroupNames label
G:=SmallGroup(432,735);
// by ID
G=gap.SmallGroup(432,735);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,-3,3,387,100,262,185,80,6060,1699,1034,201,8245,1588,223,622]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^4=f^3=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=e*c*e^-1=b*c=c*b,d*b*d^-1=c^-1,e*b*e^-1=b^-1*c,d*c*d^-1=b,c*f=f*c,e*d*e^-1=d^-1,f*d*f^-1=e,f*e*f^-1=d*e>;
// generators/relations
Export
Subgroup lattice of C2×ASL2(𝔽3) in TeX
Character table of C2×ASL2(𝔽3) in TeX