direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C2×He3, C32⋊2C6, C6.1C32, (C3×C6)⋊C3, C3.1(C3×C6), SmallGroup(54,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×He3
G = < a,b,c,d | a2=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >
Character table of C2×He3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 1 | 1 | 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 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | -1 | -1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | -1 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | -1 | -1 | -1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | linear of order 3 |
| ρ9 | 1 | -1 | 1 | 1 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | -1 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ11 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | -1 | -1 | ζ6 | ζ65 | ζ6 | ζ65 | -1 | ζ65 | -1 | ζ6 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ13 | 1 | 1 | 1 | 1 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ14 | 1 | -1 | 1 | 1 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | -1 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | linear of order 6 |
| ρ15 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ32 | 1 | ζ3 | ζ3 | -1 | -1 | ζ65 | ζ6 | ζ65 | ζ6 | -1 | ζ6 | -1 | ζ65 | linear of order 6 |
| ρ16 | 1 | -1 | 1 | 1 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | -1 | -1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | -1 | linear of order 6 |
| ρ17 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | ζ3 | 1 | ζ32 | linear of order 3 |
| ρ18 | 1 | -1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | 1 | -1 | -1 | -1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ19 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ20 | 3 | -3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3 |
| ρ22 | 3 | -3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 18 points - transitive group 18T15
Generators in S18
(1 6)(2 4)(3 5)(7 17)(8 18)(9 16)(10 14)(11 15)(12 13)
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 3 2)(4 6 5)(7 8 9)(10 12 11)(13 15 14)(16 17 18)
(1 12 16)(2 10 18)(3 11 17)(4 14 8)(5 15 7)(6 13 9)
G:=sub<Sym(18)| (1,6)(2,4)(3,5)(7,17)(8,18)(9,16)(10,14)(11,15)(12,13), (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,3,2)(4,6,5)(7,8,9)(10,12,11)(13,15,14)(16,17,18), (1,12,16)(2,10,18)(3,11,17)(4,14,8)(5,15,7)(6,13,9)>;
G:=Group( (1,6)(2,4)(3,5)(7,17)(8,18)(9,16)(10,14)(11,15)(12,13), (7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,3,2)(4,6,5)(7,8,9)(10,12,11)(13,15,14)(16,17,18), (1,12,16)(2,10,18)(3,11,17)(4,14,8)(5,15,7)(6,13,9) );
G=PermutationGroup([[(1,6),(2,4),(3,5),(7,17),(8,18),(9,16),(10,14),(11,15),(12,13)], [(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,3,2),(4,6,5),(7,8,9),(10,12,11),(13,15,14),(16,17,18)], [(1,12,16),(2,10,18),(3,11,17),(4,14,8),(5,15,7),(6,13,9)]])
G:=TransitiveGroup(18,15);
C2×He3 is a maximal subgroup of
C32⋊C12 He3⋊3C4 Q8⋊He3 D7⋊He3
C2×He3 is a maximal quotient of D7⋊He3
Matrix representation of C2×He3 ►in GL3(𝔽7) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 4 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 1 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[4,0,0,0,2,0,0,0,1],[4,0,0,0,4,0,0,0,4],[0,0,1,6,0,0,0,6,0] >;
C2×He3 in GAP, Magma, Sage, TeX
C_2\times {\rm He}_3 % in TeX
G:=Group("C2xHe3"); // GroupNames label
G:=SmallGroup(54,10);
// by ID
G=gap.SmallGroup(54,10);
# by ID
G:=PCGroup([4,-2,-3,-3,-3,150]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations
Export
Subgroup lattice of C2×He3 in TeX
Character table of C2×He3 in TeX