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Classification of finite group groups of isometries

Consider the problem of classifying the finite group groups of isometries of R^n. --For n=2 it is cyclic and dihedral groups. --For n=3 they are well known, probably from Kepler and are related to ade-classification. --For n=4 we can get them by taking the universal cover of SO(4) which is isomorphic to SU2 x SU2, though I do not know where the classification is available.

But my main question is for dimension n=5 and above. Does anybody knows the state of the art? A reference would be most helpful. Note that the finite subgroups of GLn(Z) are classified for n<=10.

Mathieu
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Classification finite group of isometries

Consider the problem of classifying the finite group of isometries of R^n. --For n=2 it is cyclic and dihedral groups. --For n=3 they are well known, probably from Kepler and are related to ade-classification. --For n=4 we can get them by taking the universal cover of SO(4) which is isomorphic to SU2 x SU2, though I do not know where the classification is available.

But my main question is for dimension n=5 and above. Does anybody knows the state of the art? A reference would be most helpful. Note that the finite subgroups of GLn(Z) are classified for n<=10.

Mathieu