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

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?

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  • $\begingroup$ 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 $(2,5)(4,7)$. $\endgroup$ – Marc Bogaerts Apr 9 '17 at 12:53

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