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Suppose that is a symmetry group. Then is one of the subgroups listed in Table 1. We say that is an untwisted subgroup of if is conjugate to a subgroup of the form where K is contained in the subgroup H given in Table 1. The untwisted symmetry groups are listed in Table 2.
It is not the case that every subgroup produces a symmetry group. For example, when ,the only symmetry group corresponding to is .(This isindependent of the restriction to untwisted symmetry groups.) To verify this point, observe that acts transitively on the cylinder .Hence if is the symmetry group of a function ,then f is the constant function.It follows that f is invariant under ,and that the symmetrysubgroup .
When contains ,the function f isconstant on each horizontal cross-section of and hence automaticallyhas the symmetry .In these cases, the only possibilities are and .Similarly, when contains then automatically and the only possibilities are and .
In all other cases, there are no restrictions on K other than thecondition .
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