non-abelian, soluble, monomial
Aliases: C32.S4, C62.5S3, C3.S4⋊C3, C3.A4⋊C6, C22⋊(C9⋊C6), C3.1(C3×S4), C32.A4⋊C2, (C2×C6).2(C3×S3), SmallGroup(216,90)
Series: Derived ►Chief ►Lower central ►Upper central
| C3.A4 — C32.S4 |
Generators and relations for C32.S4
G = < a,b,c,d,e,f | a3=b3=c2=d2=f2=1, e3=b, ab=ba, ac=ca, ad=da, eae-1=ab-1, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=b-1e2 >
Character table of C32.S4
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 12A | 12B | |
| size | 1 | 3 | 18 | 2 | 3 | 3 | 18 | 3 | 3 | 6 | 6 | 6 | 18 | 18 | 24 | 24 | 24 | 18 | 18 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ65 | ζ6 | ζ3 | 1 | ζ32 | ζ6 | ζ65 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ6 | ζ65 | ζ32 | 1 | ζ3 | ζ65 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | 0 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 2 | 0 | 0 | ζ65 | -1 | ζ6 | 0 | 0 | complex lifted from C3×S3 |
| ρ9 | 2 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | 0 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 2 | 0 | 0 | ζ6 | -1 | ζ65 | 0 | 0 | complex lifted from C3×S3 |
| ρ10 | 3 | -1 | 1 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S4 |
| ρ11 | 3 | -1 | -1 | 3 | 3 | 3 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S4 |
| ρ12 | 3 | -1 | 1 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ32 | ζ3 | 0 | 0 | 0 | ζ65 | ζ6 | complex lifted from C3×S4 |
| ρ13 | 3 | -1 | -1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | ζ32 | ζ3 | complex lifted from C3×S4 |
| ρ14 | 3 | -1 | 1 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ3 | ζ32 | 0 | 0 | 0 | ζ6 | ζ65 | complex lifted from C3×S4 |
| ρ15 | 3 | -1 | -1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | ζ3 | ζ32 | complex lifted from C3×S4 |
| ρ16 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | 4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ17 | 6 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9⋊C6 |
| ρ18 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | -2-2√-3 | -2+2√-3 | 1+√-3 | 1-√-3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | -2+2√-3 | -2-2√-3 | 1-√-3 | 1+√-3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 18 points - transitive group 18T98
Generators in S18
(2 8 5)(3 6 9)(10 16 13)(11 14 17)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 12)(2 13)(4 15)(5 16)(7 18)(8 10)
(2 13)(3 14)(5 16)(6 17)(8 10)(9 11)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 12)(2 11)(3 10)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)
G:=sub<Sym(18)| (2,8,5)(3,6,9)(10,16,13)(11,14,17), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,12)(2,13)(4,15)(5,16)(7,18)(8,10), (2,13)(3,14)(5,16)(6,17)(8,10)(9,11), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,12)(2,11)(3,10)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)>;
G:=Group( (2,8,5)(3,6,9)(10,16,13)(11,14,17), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,12)(2,13)(4,15)(5,16)(7,18)(8,10), (2,13)(3,14)(5,16)(6,17)(8,10)(9,11), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,12)(2,11)(3,10)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13) );
G=PermutationGroup([[(2,8,5),(3,6,9),(10,16,13),(11,14,17)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18)], [(1,12),(2,13),(4,15),(5,16),(7,18),(8,10)], [(2,13),(3,14),(5,16),(6,17),(8,10),(9,11)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,12),(2,11),(3,10),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13)]])
G:=TransitiveGroup(18,98);
►On 18 points - transitive group 18T101
Generators in S18
(2 8 5)(3 6 9)(10 16 13)(11 14 17)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 15)(2 16)(4 18)(5 10)(7 12)(8 13)
(2 16)(3 17)(5 10)(6 11)(8 13)(9 14)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(2 9)(3 8)(4 7)(5 6)(10 11)(12 18)(13 17)(14 16)
G:=sub<Sym(18)| (2,8,5)(3,6,9)(10,16,13)(11,14,17), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,15)(2,16)(4,18)(5,10)(7,12)(8,13), (2,16)(3,17)(5,10)(6,11)(8,13)(9,14), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (2,9)(3,8)(4,7)(5,6)(10,11)(12,18)(13,17)(14,16)>;
G:=Group( (2,8,5)(3,6,9)(10,16,13)(11,14,17), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (1,15)(2,16)(4,18)(5,10)(7,12)(8,13), (2,16)(3,17)(5,10)(6,11)(8,13)(9,14), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (2,9)(3,8)(4,7)(5,6)(10,11)(12,18)(13,17)(14,16) );
G=PermutationGroup([[(2,8,5),(3,6,9),(10,16,13),(11,14,17)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18)], [(1,15),(2,16),(4,18),(5,10),(7,12),(8,13)], [(2,16),(3,17),(5,10),(6,11),(8,13),(9,14)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(2,9),(3,8),(4,7),(5,6),(10,11),(12,18),(13,17),(14,16)]])
G:=TransitiveGroup(18,101);
C32.S4 is a maximal quotient of C32.CSU2(𝔽3) C32.GL2(𝔽3) C62.Dic3
Matrix representation of C32.S4 ►in GL6(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| -1 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | -1 | 1 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | -1 | 1 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 |
G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,-1,0,0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,-1,-1,0,0,0,0,0,1,0,0] >;
C32.S4 in GAP, Magma, Sage, TeX
C_3^2.S_4
% in TeX
G:=Group("C3^2.S4"); // GroupNames label
G:=SmallGroup(216,90);
// by ID
G=gap.SmallGroup(216,90);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,2,542,224,122,867,3244,556,1949,989]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^2=d^2=f^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^-1,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=b^-1*e^2>;
// generators/relations
Export
Subgroup lattice of C32.S4 in TeX
Character table of C32.S4 in TeX