Here is the MAGMA code to generate your group:
G:=sub<Sym(12)|(1,3,5,7,9,11)*(2,4,6,8,10,12), (2,12)*(3,11)*(4,10)*(5,9)*(6,8), (2,3)*(5,6)*(8,9)*(11,12)*(4,10)>;
I have a small function written by Tim Dokchitser that recognises direct and semi-direct products of standard groups like cyclic, symmetric, dihedral, etc. The group you have described is isomorphic to $C_2\times {\mathfrak S}_4$. The unique normal subgroup of order 2 is generated by a reflection in the central point centre of the hexagon, so its non-trivial element is given by (a,g)(b,h)(c,i)(d,j)(e,k)(f,l).
The 4-cycles are somewhat harder to visualise. One of them is (a,d,g,j)(b,c,e,f)(h,i,k,l). The copy of ${\mathfrak S}_4$ that this is contained in (there are three normal subgroups isomorphic to ${\mathfrak S}_4$) is generated by the following elements:
- (b, l)(c, k)(d, j)(e, i)(f,h) (reflection in the axis through a and g)
- (a, i, e)(b, j, f)(c, k, g)(d, l, h) (counter-clockwise rotation by $2\pi/3$)
- (a, g)(b, e)(c, f)(d, j)(h, k)(i, l) (square of the above 4-cycle)
- (a, j)(b, k)(c, i)(d, g)(e, h)(f, l) ( a weird thing somewhat similar to the previous one)
If you have more questions about this group, ask away, since magma will tell me pretty much anything I want to know.
