# Technics to find conjugacy classes of a symmetry group

I have found the following 12 symmetries for the following figure composed of a prism and two tetrahedra.

E: Identity

• E: Identity
• M: Rotation through 2PI/3 about the line crossing 1 and 8. (234)(567)
• N: Rotation through 4PI/3 about the line crossing 1 and 8. (243)(576)
• O: Rotations through Pi about line crossing midpoint of 36 and centre of opposing face (36)(45)(27)(18)
• P: Rotation through Pi about line crossing midpoing of 25 .... (37)(46)(25)(18)
• Q: Rotation ........................................of 47 ... (35)(62)(47)(18)
• R: Reflexion in the plane crossing 36 and its opposing face (24)(57)
• S: Reflexion ......................47 .......................(23)(56)
• T: Reflexion ......................25 .........................(34)(67)
• U: Reflexion in the plane crossing all vertical edges ...(36)(47)(25)(18)
• V: Rotation followed by reflexion (273546)(18)
• W: .............................. (264537)(18)

By assuming conjugates have the same cyclic structure and geometric type I found the following conjugacy classes:

• {E}
• {M,N}
• {O,P,Q}
• {R,S,T}
• {U}
• {V,W}

Are these correct conjugacy classes of the set of symmetries? Any easier technic to verify my results?

How do I find normal subgroups of order 4 using the results above?

• The group you describe is in fact the symmetric group $S_4$. The only normal subgroup of order $4$ is the Klein Vierergruppe $C_2 \times C_2$ generated by $(3,6)(4,7)$ and $(2,5)(4,7)$. – Marc Bogaerts Apr 9 '17 at 12:53