The 48 isometries of the left-handed die

and

the regular tetrahedron as a subgroup of the cube





-- The identity


-- One inversion: (16)(25)(34)


-- Three 4-edge reflections: (16), (25), (34)


-- Six 4-vertex reflections: (12)(56), (15)(26), (14)(36), (13)(46), (24)(35), (23)(45)


-- Three twofold 2-side rotations: (25)(34), (16)(34), (16)(25)


-- Six twofold 2-edge rotations: (16)(24)(35), (16)(23)(45), (14)(25)(36), (13)(25)(46), (12)(34)(56), (15)(26)(34)


-- Six fourfold rotations: (2453), (2354), (1463), (1364), (1265), (1562)


-- Eight threefold rotations: (145)(263), (154)(362), (124)(365), (142)(563), (123)(465), (132)(564), (135)(264), (153)(462)


-- Six fourfold 4-edge rotoreflections: (16)(2453), (16)(2354), (25)(1463), (25)(1364), (34)(1265), (34)(1562)


-- Eight sixfold rotoreflections: (124653), (135642), (132645), (154623), (142635), (153624), (123654), (145632)



I have a question for you: why don't you use sides instead of vertices? -- Bonita Bryson, December 2004