In terms of references, it seems there are a number references about the orientation preserving subgroups of these groups and their corresponding 3-dimension orbifolds.

Here are some:

Maclachlan and Reid's "The Arithmetic of Hyperbolic 3-Manifolds" has the hyperbolic tetrahedral groups arranged into commensurability classes in the Appendix (Chapter 13) and a longer discussion of about the groups in section 4.7.

Representations of the orientation preserving subgroups of these Coxeter groups into $PSL(2,\mathbb{C})$ as computed by Grant Lakeland are available here. (Notes titled "Matrix Realizations of the Hyperbolic Tetrahedral Groups in ${PSL}_2(\mathbb C)$.")

The non-hyperbolic examples of these Coxeter groups are classified according to the geometries of their quotients in William Dunbar's paper "Geometric Orbifolds."

In terms of references, it seems there are a number references about the orientation preserving subgroups of these groups and their corresponding 3-dimension orbifolds.

Here are some:

Maclachlan and Reid's "The Arithmetic of Hyperbolic 3-Manifolds" has the hyperbolic tetrahedral groups arranged into commensurability classes in the Appendix (Chapter 13) and a longer discussion of about the groups in section 4.7.

Representations of the orientation preserving subgroups of these Coxeter groups into $PSL(2,\mathbb{C})$ as computed by Grant Lakeland are available in Matrix Realizations of the Hyperbolic Tetrahedral Groups in ${PSL}_2(\mathbb C)$.

The non-hyperbolic examples of these Coxeter groups are classified according to the geometries of their quotients in William Dunbar's paper "Geometric Orbifolds."

In terms of references, it seems there are a number references about the orientation preserving subgroups of these groups and their corresponding 3-dimension orbifolds.

Here are some:

Maclachlan and Reid's "The Arithmetic of Hyperbolic 3-manifolds" has the hyperbolic tetrahedral groups arranged into commensurability classes in the Appendix (Chapter 13) and a longer discussion of about the groups in section 4.7.

Representations of the orientation preserving subgroups of these Coxeter groups into $PSL(2,\mathbb{C})$ as computed by Grant Lakeland are available here.

The non-hyperbolic examples of these Coxeter groups are classified according to the geometries of their quotients in William Dunbar's paper "Geometric Orbifolds."

In terms of references, it seems there are a number references about the orientation preserving subgroups of these groups and their corresponding 3-dimension orbifolds.

Here are some:

Maclachlan and Reid's "The Arithmetic of Hyperbolic 3-Manifolds" has the hyperbolic tetrahedral groups arranged into commensurability classes in the Appendix (Chapter 13) and a longer discussion of about the groups in section 4.7.

Representations of the orientation preserving subgroups of these Coxeter groups into $PSL(2,\mathbb{C})$ as computed by Grant Lakeland are available here. (Notes titled "Matrix Realizations of the Hyperbolic Tetrahedral Groups in ${PSL}_2(\mathbb C)$.")

The non-hyperbolic examples of these Coxeter groups are classified according to the geometries of their quotients in William Dunbar's paper "Geometric Orbifolds."