Supergroups for six series of hyperbolic simplex groups

There are investigated supergroups of some hyperbolic space groups with simplicial fundamental domain. Six simplices considered here from 9] are collected in families F9 (T ₂₃, T ₆₄), F10 (T ₂₁, T ₄₉, T ₆₁), F29 (T ₃₄). All of them have the same symmetry by half-turn h, with axis through the midpoints of edges A ₀ A ₁ and A ₂ A ₃. Since that isometry identifies pairs of points, if a supergroup with such smaller fundamental domain exists, it is of index 2. At the side pairings of T ₃₄ this half-turn implies additional reflections, equal parameters 2a = 6b, and leads to Family 2, considered in 9]. Other possibility to find supergroups is when the simplices have vertices out of the absolute. In that case we can truncate them by polar planes of the vertices and the new polyhedra are fundamental domains of richer groups.