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1. Surprisingly, I found explicit lists of discrete subgroups of the orthogonal group O(n) for up to n=8 dimensions on the wikipedia page for point groups, with rather unspecific references, however. Point groups is another name for discrete subgroups of O(n). [UPDATE+CORRECTION: For dimensions n=4 and larger, only the point groups which are generated by reflections (Coxeter groups) are listed. In particularly, subgroups of SO(n) (which include no matrix of determinant $-$1) are missing.]
2. There is an old sequence of two long papers by Threlfall and Seifert, part I Mathematische Annalen 1931, Volume 104, Issue 1, pp. 1-70, part II 1933, Volume 107, Issue 1, pp. 543-586, where they apparently do the classification of discrete subgroups of SO(4) by associating to each element of SO(4) a pair of rotations from SO(3). (Although my native language is German, I had a hard time reading (through) this, because I am not used to the terminology that was used at that time.) [Addition: These results are mentioned in the book by Conway and Smith on quaternions and octonions; Conway and Smith say that the list is complete, but contains duplicates.]
3. I have a rather wild conjecture (true up to three dimensions). [UPDATE 2: wrong in 4 dimensions]

Every discrete point group in n dimensions is the symmetry group of an n-dimensional polytope which is the Cartesian product of regular polytopes, or a subgroup thereof.

[UPDATE 2: One counterexample in 4D is the group $\pm [I\times C_n]$ in Table 4.1 of Conway and SteinSloane's book, p. 44, for sufficiently high $n$. It is isomorphic to a subgroup of a direct product of lower-dimensional point groups, in the group-theoretic sense, but geometrically, it is not the symmetry group of a Cartesian product of two objects in orthogonal subspaces. An orbit $A$ of a point under this group can be constructed as follows. Consider the Hopf fibration $f\colon S^3\to S^2$, and take the twelve great circles which are the pre-images of the vertices of a regular icosahedron on $S^2$. The point set $A$ consists of 12 regular $n$-gons inscribed in these circles. Probably, the first group in the list, $\pm [I\times O]$, is also a counterexample, since it is not contained in an achiral group.]

[UPDATE 1: Norman Johnson pointed out counterexamples: The symmetries of the root lattices E6, E7, E8 in 6, 7, and 8 dimensions. (I could not yet fully convinced myself that they are indeed counterexamples.) So dimensions 4 and 5 remain open. If I extend my conjecture to include the polytopes which have those E6, E7, or E8 symmetries, in addition to the regular polytopes, in which dimension would the next counterexamples be?]

For example, the symmetries of an $m$-gonal anti-prism in 3-space are contained in the symmetries of the $2m$-sided prism, which is the 1-simplex $\times$ the regular $2m$-gon.

Since the regular polytopes are known in all dimensions, this would give an easy way to obtain all finite point groups. (at least in principle).

1. Surprisingly, I found explicit lists of discrete subgroups of the orthogonal group O(n) for up to n=8 dimensions on the wikipedia page for point groups, with rather unspecific references, however. Point groups is another name for discrete subgroups of O(n). [UPDATE+CORRECTION: For dimensions n=4 and larger, only the point groups which are generated by reflections (Coxeter groups) are listed. In particularly, subgroups of SO(n) (which include no matrix of determinant $-$1) are missing.]
2. There is an old sequence of two long papers by Threlfall and Seifert, part I Mathematische Annalen 1931, Volume 104, Issue 1, pp. 1-70, part II 1933, Volume 107, Issue 1, pp. 543-586, where they apparently do the classification of discrete subgroups of SO(4) by associating to each element of SO(4) a pair of rotations from SO(3). (Although my native language is German, I had a hard time reading (through) this, because I am not used to the terminology that was used at that time.) [Addition: These results are mentioned in the book by Conway and Smith on quaternions and octonions; Conway and Smith say that the list is complete, but contains duplicates.]
3. I have a rather wild conjecture (true up to three dimensions). [UPDATE 2: wrong in 4 dimensions]

Every discrete point group in n dimensions is the symmetry group of an n-dimensional polytope which is the Cartesian product of regular polytopes, or a subgroup thereof.

[UPDATE 2: One counterexample in 4D is the group $\pm [I\times C_n]$ in Table 4.1 of Conway and SmithSloane's book, p. 44, for sufficiently high $n$. It is isomorphic to a subgroup of a direct product of lower-dimensional point groups, in the group-theoretic sense, but geometrically, it is not the symmetry group of a Cartesian product of two objects in orthogonal subspaces. An orbit $A$ of a point under this group can be constructed as follows. Consider the Hopf fibration $f\colon S^3\to S^2$, and take the twelve great circles which are the pre-images of the vertices of a regular icosahedron on $S^2$. The point set $A$ consists of 12 regular $n$-gons inscribed in these circles. Probably, the first group in the list, $\pm [I\times O]$, is also a counterexample, since it is not contained in an achiral group.]

[UPDATE 1: Norman Johnson pointed out counterexamples: The symmetries of the root lattices E6, E7, E8 in 6, 7, and 8 dimensions. (I could not yet fully convinced myself that they are indeed counterexamples.) So dimensions 4 and 5 remain open. If I extend my conjecture to include the polytopes which have those E6, E7, or E8 symmetries, in addition to the regular polytopes, in which dimension would the next counterexamples be?]

For example, the symmetries of an $m$-gonal anti-prism in 3-space are contained in the symmetries of the $2m$-sided prism, which is the 1-simplex $\times$ the regular $2m$-gon.

Since the regular polytopes are known in all dimensions, this would give an easy way to obtain all finite point groups. (at least in principle).

1. Surprisingly, I found explicit lists of discrete subgroups of the orthogonal group O(n) for up to n=8 dimensions on the wikipedia page for point groups, with rather unspecific references, however. Point groups is another name for discrete subgroups of O(n). [UPDATE+CORRECTION: For dimensions n=4 and larger, only the point groups which are generated by reflections (Coxeter groups) are listed. In particularly, subgroups of SO(n) (which include no matrix of determinant $-$1) are missing.]
2. There is an old sequence of two long papers by Threlfall and Seifert, part I Mathematische Annalen 1931, Volume 104, Issue 1, pp. 1-70, part II 1933, Volume 107, Issue 1, pp. 543-586, where they apparently do the classification of discrete subgroups of SO(4) by associating to each element of SO(4) a pair of rotations from SO(3). (Although my native language is German, I had a hard time reading (through) this, because I am not used to the terminology that was used at that time.) [Addition: These results are mentioned in the book by Conway and Smith on quaternions and octonions; Conway and Smith say that the list is complete, but contains duplicates.]
3. I have a rather wild conjecture (true up to three dimensions). [UPDATE 2: wrong in 4 dimensions]

Every discrete point group in n dimensions is the symmetry group of an n-dimensional polytope which is the Cartesian product of regular polytopes, or a subgroup thereof.

[UPDATE 2: One counterexample in 4D is the group $\pm [I\times C_n]$ in Table 4.1 of Conway and SteinSloane's book, p. 44, for sufficiently high $n$. It is isomorphic to a subgroup of a direct product of lower-dimensional point groups, in the group-theoretic sense, but geometrically, it is not the symmetry group of a Cartesian product of two objects in orthogonal subspaces. An orbit $A$ of a point under this group can be constructed as follows. Consider the Hopf fibration $f\colon S^4\to S^2$, and take the twelve great circles which are the pre-images of the vertices of a regular icosahedron on $S^2$. The point set $A$ consists of 12 regular $n$-gons inscribed in these circles. Probably, the first group in the list, $\pm [I\times O]$, is also a counterexample, since it is not contained in an achiral group.]

[UPDATE 1: Norman Johnson pointed out counterexamples: The symmetries of the root lattices E6, E7, E8 in 6, 7, and 8 dimensions. (I could not yet fully convinced myself that they are indeed counterexamples.) So dimensions 4 and 5 remain open. If I extend my conjecture to include the polytopes which have those E6, E7, or E8 symmetries, in addition to the regular polytopes, in which dimension would the next counterexamples be?]

For example, the symmetries of an $m$-gonal anti-prism in 3-space are contained in the symmetries of the $2m$-sided prism, which is the 1-simplex $\times$ the regular $2m$-gon.

Since the regular polytopes are known in all dimensions, this would give an easy way to obtain all finite point groups. (at least in principle).

1. Surprisingly, I found explicit lists of discrete subgroups of the orthogonal group O(n) for up to n=8 dimensions on the wikipedia page for point groups, with rather unspecific references, however. Point groups is another name for discrete subgroups of O(n). [UPDATE+CORRECTION: For dimensions n=4 and larger, only the point groups which are generated by reflections (Coxeter groups) are listed. In particularly, subgroups of SO(n) (which include no matrix of determinant $-$1) are missing.]
2. There is an old sequence of two long papers by Threlfall and Seifert, part I Mathematische Annalen 1931, Volume 104, Issue 1, pp. 1-70, part II 1933, Volume 107, Issue 1, pp. 543-586, where they apparently do the classification of discrete subgroups of SO(4) by associating to each element of SO(4) a pair of rotations from SO(3). (Although my native language is German, I had a hard time reading (through) this, because I am not used to the terminology that was used at that time.) [Addition: These results are mentioned in the book by Conway and Smith on quaternions and octonions; Conway and Smith say that the list is complete, but contains duplicates.]
3. I have a rather wild conjecture (true up to three dimensions). [UPDATE 2: wrong in 4 dimensions]

Every discrete point group in n dimensions is the symmetry group of an n-dimensional polytope which is the Cartesian product of regular polytopes, or a subgroup thereof.

[UPDATE 2: One counterexample in 4D is the group $\pm [I\times C_n]$ in Table 4.1 of Conway and SteinSloane's book, p. 44, for sufficiently high $n$. It is isomorphic to a subgroup of a direct product of lower-dimensional point groups, in the group-theoretic sense, but geometrically, it is not the symmetry group of a Cartesian product of two objects in orthogonal subspaces. An orbit $A$ of a point under this group can be constructed as follows. Consider the Hopf fibration $f\colon S^3\to S^2$, and take the twelve great circles which are the pre-images of the vertices of a regular icosahedron on $S^2$. The point set $A$ consists of 12 regular $n$-gons inscribed in these circles. Probably, the first group in the list, $\pm [I\times O]$, is also a counterexample, since it is not contained in an achiral group.]

[UPDATE 1: Norman Johnson pointed out counterexamples: The symmetries of the root lattices E6, E7, E8 in 6, 7, and 8 dimensions. (I could not yet fully convinced myself that they are indeed counterexamples.) So dimensions 4 and 5 remain open. If I extend my conjecture to include the polytopes which have those E6, E7, or E8 symmetries, in addition to the regular polytopes, in which dimension would the next counterexamples be?]

For example, the symmetries of an $m$-gonal anti-prism in 3-space are contained in the symmetries of the $2m$-sided prism, which is the 1-simplex $\times$ the regular $2m$-gon.

Since the regular polytopes are known in all dimensions, this would give an easy way to obtain all finite point groups. (at least in principle).

1. Surprisingly, I found explicit lists of discrete subgroups of the orthogonal group O(n) for up to n=8 dimensions on the wikipedia page for point groups, with rather unspecific references, however. Point groups is another name for discrete subgroups of O(n). [UPDATE+CORRECTION: For dimensions n=4 and larger, only the point groups which are generated by reflections (Coxeter groups) are listed. In particularly, subgroups of SO(n) (which include no matrix of determinant $-$1) are missing.]
2. There is an old sequence of two long papers by Threlfall and Seifert, part I Mathematische Annalen 1931, Volume 104, Issue 1, pp. 1-70, part II 1933, Volume 107, Issue 1, pp. 543-586, where they apparently do the classification of discrete subgroups of SO(4) by associating to each element of SO(4) a pair of rotations from SO(3). (Although my native language is German, I had a hard time reading (through) this, because I am not used to the terminology that was used at that time.) [Addition: These results are mentioned in the book by Conway and Smith on quaternions and octonions; Conway and Smith say that the list is complete, but contains duplicates.]
3. I have a rather wild conjecture (true up to three dimensions). [UPDATE 2: wrong in 4 dimensions]

Every discrete point group in n dimensions is the symmetry group of an n-dimensional polytope which is the Cartesian product of regular polytopes, or a subgroup thereof.

[UPDATE 2: One counterexample in 4D is the group $\pm [I\times C_n]$ in Table 4.1 of Conway and Sloane's book, p. 44, for sufficiently high $n$. It is isomorphic to a subgroup of a direct product of lower-dimensional point groups, in the group-theoretic sense, but geometrically, it is not the symmetry group of a Cartesian product of two objects in orthogonal subspaces. An orbit $A$ of a point under this group can be constructed as follows. Consider the Hopf fibration $f\colon S^3\to S^2$, and take the twelve great circles which are the pre-images of the vertices of a regular icosahedron on $S^2$. The point set $A$ consists of 12 regular $n$-gons inscribed in these circles. Probably, the first group in the list, $\pm [I\times O]$, is also a counterexample, since it is not contained in an achiral group.]

[UPDATE 1: Norman Johnson pointed out counterexamples: The symmetries of the root lattices E6, E7, E8 in 6, 7, and 8 dimensions. (I could not yet fully convinced myself that they are indeed counterexamples.) So dimensions 4 and 5 remain open. If I extend my conjecture to include the polytopes which have those E6, E7, or E8 symmetries, in addition to the regular polytopes, in which dimension would the next counterexamples be?]

For example, the symmetries of an $m$-gonal anti-prism in 3-space are contained in the symmetries of the $2m$-sided prism, which is the 1-simplex $\times$ the regular $2m$-gon.

Since the regular polytopes are known in all dimensions, this would give an easy way to obtain all finite point groups. (at least in principle).

1. Surprisingly, I found explicit lists of discrete subgroups of the orthogonal group O(n) for up to n=8 dimensions on the wikipedia page for point groups, with rather unspecific references, however. Point groups is another name for discrete subgroups of O(n). [UPDATE+CORRECTION: For dimensions n=4 and larger, only the point groups which are generated by reflections (Coxeter groups) are listed. In particularly, subgroups of SO(n) (which include no matrix of determinant $-$1) are missing.]
2. There is an old sequence of two long papers by Threlfall and Seifert, part I Mathematische Annalen 1931, Volume 104, Issue 1, pp. 1-70, part II 1933, Volume 107, Issue 1, pp. 543-586, where they apparently do the classification of discrete subgroups of SO(4) by associating to each element of SO(4) a pair of rotations from SO(3). (Although my native language is German, I had a hard time reading (through) this, because I am not used to the terminology that was used at that time.) [Addition: These results are mentioned in the book by Conway and Smith on quaternions and octonions; Conway and Smith say that the list is complete, but contains duplicates.]
3. I have a rather wild conjecture (true up to three dimensions). [UPDATE 2: wrong in 4 dimensions]

Every discrete point group in n dimensions is the symmetry group of an n-dimensional polytope which is the Cartesian product of regular polytopes, or a subgroup thereof.

[UPDATE 2: One counterexample in 4D is the group $\pm [I\times C_n]$ in Table 4.1 of Conway and SteinSloane's book, p. 44, for sufficiently high $n$. It is isomorphic to a subgroup of a direct product of lower-dimensional point groups, in the group-theoretic sense, but geometrically, it is not the symmetry group of a Cartesian product of two objects in orthogonal subspaces. An orbit $A$ of a point under this group can be constructed as follows. Consider the Hopf fibration $f\colon S^4\to S^2$, and take the twelve great circles which are the pre-images of the vertices of a regular icosahedron on $S^2$. The point set $A$ consists of 12 regular $n$-gons inscribed in these circles. Probably, the first group in the list, $\pm [I\times O]$, is also a counterexample, since it is not contained in an achiral group.]

[UPDATE 1: Norman Johnson pointed out counterexamples: The symmetries of the root lattices E6, E7, E8 in 6, 7, and 8 dimensions. (I could not yet fully convinced myself that they are indeed counterexamples.) So dimensions 4 and 5 remain open. If I extend my conjecture to include the polytopes which have those E6, E7, or E8 symmetries, in addition to the regular polytopes, in which dimension would the next counterexamples be?]

For example, the symmetries of an $m$-gonal anti-prism in 3-space are contained in the symmetries of the $2m$-sided prism, which is the 1-simplex $\times$ the regular $2m$-gon.

Since the regular polytopes are known in all dimensions, this would give an easy way to obtain all finite point groups. (at least in principle).