Uniform polyhedron compound

In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts transitively on the compound's vertices.
The uniform polyhedron compounds were first enumerated by John Skilling in 1976, with a proof that the enumeration is complete. The following table lists them according to his numbering.
The prismatic compounds of prisms exist only when, and when and are coprime. The uniform prismatic compounds of antiprisms exist only when, and when and are coprime. Furthermore, when, the antiprisms degenerate into tetrahedra with digonal bases.

Compound	Bowers acronym	Picture	Polyhedral count	Polyhedral type	Faces	Edges	Vertices	Notes	Symmetry group	Subgroup restricting to one constituent
UC₀₁	sis	50px	6	tetrahedra	24	36	24	Rotational freedom	T_d	S₄
UC₀₂	dis	50px	12	tetrahedra	48	72	48	Rotational freedom	O_h	S₄
UC₀₃	snu	50px	6	tetrahedra	24	36	24		O_h	D_2d
UC₀₄	so	50px	2	tetrahedra	8	12	8	Regular	O_h	T_d
UC₀₅	ki	50px	5	tetrahedra	20	30	20	Regular	I	T
UC₀₆	e	50px	10	tetrahedra	40	60	20	Regular 2 polyhedra per vertex	I_h	T
UC₀₇	risdoh	50px	6	cubes		72	48	Rotational freedom	O_h	C_4h
UC₀₈	rah	50px	3	cubes		36	24		O_h	D_4h
UC₀₉	rhom	50px	5	cubes	30	60	20	Regular 2 polyhedra per vertex	I_h	T_h
UC₁₀	dissit	50px	4	octahedra		48	24	Rotational freedom	T_h	S₆
UC₁₁	daso	50px	8	octahedra		96	48	Rotational freedom	O_h	S₆
UC₁₂	sno	50px	4	octahedra		48	24		O_h	D_3d
UC₁₃	addasi	50px	20	octahedra		240	120	Rotational freedom	I_h	S₆
UC₁₄	dasi	50px	20	octahedra		240	60	2 polyhedra per vertex	I_h	S₆
UC₁₅	gissi	50px	10	octahedra		120	60		I_h	D_3d
UC₁₆	si	50px	10	octahedra		120	60		I_h	D_3d
UC₁₇	se	50px	5	octahedra	40	60	30	Regular	I_h	T_h
UC₁₈	hirki	50px	5	tetrahemihexahedra	20 15	60	30		I	T
UC₁₉	sapisseri	50px	20	tetrahemihexahedra	60	240	60	2 polyhedra per vertex	I	C₃
UC₂₀	-	50px	2n	p/''q-gonal prisms	4n'' 2np	6np	4np	Rotational freedom	D_nph	C_ph
UC₂₁	-	50px	n	p/''q-gonal prisms	2n'' np	3np	2np		D_nph	D_ph
UC₂₂	-	50px	2n	p/''q-gonal antiprisms	4n'' 4np	8np	4np	Rotational freedom	D_npd D_nph	S_2p
UC₂₃	-	50px	n	p/''q-gonal antiprisms	2n'' 2np	4np	2np		D_npd D_nph	D_pd
UC₂₄	-	50px	2n	p/''q-gonal antiprisms	4n'' 4np	8np	4np	Rotational freedom	D_nph	C_ph
UC₂₅	-	50px	n	p/''q-gonal antiprisms	2n'' 2np	4np	2np		D_nph	D_ph
UC₂₆	gadsid	50px	12	pentagonal antiprisms	120 24	240	120	Rotational freedom	I_h	S₁₀
UC₂₇	gassid	50px	6	pentagonal antiprisms	60 12	120	60		I_h	D_5d
UC₂₈	gidasid	50px	12	pentagrammic crossed antiprisms	120 24	240	120	Rotational freedom	I_h	S₁₀
UC₂₉	gissed	50px	6	pentagrammic crossed antiprisms	60 12	120	60		I_h	D_5d
UC₃₀	ro	50px	4	triangular prisms	8 12	36	24		O	D₃
UC₃₁	dro	50px	8	triangular prisms	16 24	72	48		O_h	D₃
UC₃₂	kri	50px	10	triangular prisms	20 30	90	60		I	D₃
UC₃₃	dri	50px	20	triangular prisms	40 60	180	60	2 polyhedra per vertex	I_h	D₃
UC₃₄	kred	50px	6	pentagonal prisms	30 12	90	60		I	D₅
UC₃₅	dird	50px	12	pentagonal prisms	60 24	180	60	2 polyhedra per vertex	I_h	D₅
UC₃₆	gikrid	50px	6	pentagrammic prisms	30 12	90	60		I	D₅
UC₃₇	giddird	50px	12	pentagrammic prisms	60 24	180	60	2 polyhedra per vertex	I_h	D₅
UC₃₈	griso	50px	4	hexagonal prisms	24 8	72	48		O_h	D_3d
UC₃₉	rosi	50px	10	hexagonal prisms	60 20	180	120		I_h	D_3d
UC₄₀	rassid	50px	6	decagonal prisms	60 12	180	120		I_h	D_5d
UC₄₁	grassid	50px	6	decagrammic prisms	60 12	180	120		I_h	D_5d
UC₄₂	gassic	50px	3	square antiprisms	24 6	48	24		O	D₄
UC₄₃	gidsac	50px	6	square antiprisms	48 12	96	48		O_h	D₄
UC₄₄	sassid	50px	6	pentagrammic antiprisms	60 12	120	60		I	D₅
UC₄₅	sadsid	50px	12	pentagrammic antiprisms	120 24	240	120		I_h	D₅
UC₄₆	siddo	50px	2	icosahedra		60	24		O_h	T_h
UC₄₇	sne	50px	5	icosahedra		150	60		I_h	T_h
UC₄₈	presipsido	50px	2	great dodecahedra	24	60	24		O_h	T_h
UC₄₉	presipsi	50px	5	great dodecahedra	60	150	60		I_h	T_h
UC₅₀	passipsido	50px	2	small stellated dodecahedra	24	60	24		O_h	T_h
UC₅₁	passipsi	50px	5	small stellated dodecahedra	60	150	60		I_h	T_h
UC₅₂	sirsido	50px	2	great icosahedra		60	24		O_h	T_h
UC₅₃	sirsei	50px	5	great icosahedra		150	60		I_h	T_h
UC₅₄	tisso	50px	2	truncated tetrahedra	8 8	36	24		O_h	T_d
UC₅₅	taki	50px	5	truncated tetrahedra	20 20	90	60		I	T
UC₅₆	te	50px	10	truncated tetrahedra	40 40	180	120		I_h	T
UC₅₇	tar	50px	5	truncated cubes	40 30	180	120		I_h	T_h
UC₅₈	quitar	50px	5	stellated truncated hexahedra	40 30	180	120		I_h	T_h
UC₅₉	arie	50px	5	cuboctahedra	40 30	120	60		I_h	T_h
UC₆₀	gari	50px	5	cubohemioctahedra	30 20	120	60		I_h	T_h
UC₆₁	iddei	50px	5	octahemioctahedra	40 20	120	60		I_h	T_h
UC₆₂	rasseri	50px	5	rhombicuboctahedra	40	240	120		I_h	T_h
UC₆₃	rasher	50px	5	small rhombihexahedra	60 30	240	120		I_h	T_h
UC₆₄	rahrie	50px	5	small cubicuboctahedra	40 30 30	240	120		I_h	T_h
UC₆₅	raquahri	50px	5	great cubicuboctahedra	40 30 30	240	120		I_h	T_h
UC₆₆	rasquahr	50px	5	great rhombihexahedra	60 30	240	120		I_h	T_h
UC₆₇	rosaqri	50px	5	nonconvex great rhombicuboctahedra	40	240	120		I_h	T_h
UC₆₈	disco	50px	2	snub cubes	12	120	48		O_h	O
UC₆₉	dissid	50px	2	snub dodecahedra	24	300	120		I_h	I
UC₇₀	giddasid	50px	2	great snub icosidodecahedra	24	300	120		I_h	I
UC₇₁	gidsid	50px	2	great inverted snub icosidodecahedra	24	300	120		I_h	I
UC₇₂	gidrissid	50px	2	great retrosnub icosidodecahedra	24	300	120		I_h	I
UC₇₃	disdid	50px	2	snub dodecadodecahedra	120 24 24	300	120		I_h	I
UC₇₄	idisdid	50px	2	inverted snub dodecadodecahedra	120 24 24	300	120		I_h	I
UC₇₅	desided	50px	2	snub icosidodecadodecahedra	24 24	360	120		I_h	I