direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C2×3- 1+2, C18⋊C3, C9⋊2C6, C32.C6, C6.2C32, (C3×C6).C3, C3.2(C3×C6), SmallGroup(54,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×3- 1+2
G = < a,b,c | a2=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Character table of C2×3- 1+2
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 9A | 9B | 9C | 9D | 9E | 9F | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ65 | ζ6 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | linear of order 3 |
| ρ7 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ65 | ζ6 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ6 | ζ65 | -1 | ζ65 | -1 | ζ6 | linear of order 6 |
| ρ8 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ65 | ζ6 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | -1 | linear of order 6 |
| ρ9 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ6 | ζ65 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ65 | ζ6 | -1 | ζ6 | -1 | ζ65 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | linear of order 3 |
| ρ11 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ12 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ13 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ14 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | linear of order 3 |
| ρ15 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ6 | ζ65 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | linear of order 6 |
| ρ16 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ17 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ6 | ζ65 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | -1 | linear of order 6 |
| ρ18 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ19 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from 3- 1+2 |
| ρ22 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from 3- 1+2 |
►On 18 points - transitive group 18T14
Generators in S18
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(2 8 5)(3 6 9)(10 13 16)(12 18 15)
G:=sub<Sym(18)| (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (2,8,5)(3,6,9)(10,13,16)(12,18,15)>;
G:=Group( (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (2,8,5)(3,6,9)(10,13,16)(12,18,15) );
G=PermutationGroup([[(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15)]])
G:=TransitiveGroup(18,14);
C2×3- 1+2 is a maximal subgroup of
C9⋊C12 C18.A4 Q8⋊3- 1+2 C9⋊3F7 C9⋊4F7 C32.F7
C2×3- 1+2 is a maximal quotient of C9⋊3F7 C9⋊4F7 C32.F7
Matrix representation of C2×3- 1+2 ►in GL3(𝔽7) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 0 | 0 | 1 |
| 5 | 0 | 0 |
| 0 | 5 | 0 |
| 2 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[0,5,0,0,0,5,1,0,0],[2,0,0,0,4,0,0,0,1] >;
C2×3- 1+2 in GAP, Magma, Sage, TeX
C_2\times 3_-^{1+2} % in TeX
G:=Group("C2xES-(3,1)"); // GroupNames label
G:=SmallGroup(54,11);
// by ID
G=gap.SmallGroup(54,11);
# by ID
G:=PCGroup([4,-2,-3,-3,-3,77,150]);
// Polycyclic
G:=Group<a,b,c|a^2=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
Export
Subgroup lattice of C2×3- 1+2 in TeX
Character table of C2×3- 1+2 in TeX