The Regge group is the finite group 
of linear transformations
of the six edge lengths, or equivalently the six dihedral
angles, of a labeled tetrahedron generated by
the 24 vertex relabelings and the three Regge
transformations. Any one Regge transformation together with the relabelings generates
the full group because the three transformations are conjugate under relabeling.
Abstractly,
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Although 
is a direct product, the geometric 
subgroup of vertex relabelings is not one of its direct
factors. The group orbit of a labeled tetrahedron
under 
is called its Regge orbit. Regge orbits
organize tetrahedra related by the Regge symmetry,
including the sporadic rational tetrahedra
(Kedlaya et al. 2020).
The group originated in Regge's additional symmetries of Racah W-coefficients, equivalently Wigner 6j-symbols (Regge 1959).
