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