There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.
John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts.
The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.
There are four infinite cyclic symmetry families, with n = 2 or higher.
O(1) is a single orthogonal reflection, dihedral symmetry order 2, Dih1.