An order of convex polyhedras whose faces are regular polygons.
 

Reference: [URL="https://oeis.org/A180916"]https://oeis.org/A180916[/URL]

Like [URL="https://en.wikipedia.org/wiki/List_of_Johnson_solids"]Johnson solids[/URL], such convex polyhedras (including: [URL="https://en.wikipedia.org/wiki/Platonic_solid"]Platonic solids[/URL], [URL="https://en.wikipedia.org/wiki/Prism_(geometry)"]uniform prisms[/URL], [URL="https://en.wikipedia.org/wiki/Antiprism"]uniform antiprisms[/URL], [URL="https://en.wikipedia.org/wiki/Archimedean_solid"]Archimedean solids[/URL], and [URL="https://en.wikipedia.org/wiki/Johnson_solid"]Johnson solids[/URL]), although there are infinitely many such convex polyhedras, there is a system to sort them:

1. By number of faces.
2. If the number of faces are same, then: Platonic solid --> prism --> antiprism --> Archimedean solid --> Johnson solid
3. If two or more Archimedean solids have the same number of faces, then sorted by [URL="https://en.wikipedia.org/wiki/Archimedean_solid#Classification"]the order of Archimedean solids in Wikipedia[/URL], if two or more Johnson solids have the same number of faces, then sorted by the indices in Johnson's paper.

Since the convex polyhedras with given number of faces is already finite, the number of faces is already can be the main order of the convex polyhedras, I do not think it is necessary to use the number of edges or the number of vertices to sort (of course, for a given number of faces, these two orders are the same, since F+V-E=2)

Such order will be:

1. [URL="https://en.wikipedia.org/wiki/Regular_tetrahedron"]Regular tetrahedron[/URL] (the Platonic solid with 4 faces) (f=4)
2. [URL="https://en.wikipedia.org/wiki/Triangular_prism"]Uniform triangular prism[/URL] (the uniform prism with 3-gonal) (f=5)
3. [URL="https://en.wikipedia.org/wiki/Square_pyramid"]Equilateral square pyramid[/URL] (the Johnson solid J1) (f=5)
4. [URL="https://en.wikipedia.org/wiki/Cube"]Cube[/URL] (the Platonic solid with 6 faces) (the uniform prism with 4-gonal) (f=6)
5. [URL="https://en.wikipedia.org/wiki/Pentagonal_pyramid"]Pentagonal pyramid[/URL] (the Johnson solid J2) (f=6)
6. [URL="https://en.wikipedia.org/wiki/Triangular_bipyramid"]Triangular bipyramid[/URL] (the Johnson solid J12) (f=6)
7. [URL="https://en.wikipedia.org/wiki/Pentagonal_prism"]Uniform pentagonal prism[/URL] (the uniform prism with 5-gonal) (f=7)
8. [URL="https://en.wikipedia.org/wiki/Elongated_triangular_pyramid"]Elongated triangular pyramid[/URL] (the Johnson solid J7) (f=7)
9. [URL="https://en.wikipedia.org/wiki/Regular_octahedron"]Regular octahedron[/URL] (the Platonic solid with 8 faces) (the uniform antiprism with 3-gonal) (f=8)
10. [URL="https://en.wikipedia.org/wiki/Hexagonal_prism"]Hexagonal prism[/URL] (the uniform prism with 6-gonal) (f=8)
11. [URL="https://en.wikipedia.org/wiki/Truncated_tetrahedron"]Truncated tetrahedron[/URL] (an Archimedean solid) (f=8)
12. [URL="https://en.wikipedia.org/wiki/Triangular_cupola"]Triangular cupola[/URL] (the Johnson solid J3) (f=8)
13. [URL="https://en.wikipedia.org/wiki/Gyrobifastigium"]Gyrobifastigium[/URL] (the Johnson solid J26) (f=8)
14. [URL="https://en.wikipedia.org/wiki/Augmented_triangular_prism"]Augmented triangular prism[/URL] (the Johnson solid J49) (f=8)
15. [URL="https://en.wikipedia.org/wiki/Tridiminished_icosahedron"]Tridiminished icosahedron[/URL] (the Johnson solid J63) (f=8)
16. [URL="https://en.wikipedia.org/wiki/Heptagonal_prism"]Heptagonal prism[/URL] (the uniform prism with 7-gonal) (f=9)
17. [URL="https://en.wikipedia.org/wiki/Elongated_square_pyramid"]Elongated square pyramid[/URL] (the Johnson solid J8) (f=9)
18. [URL="https://en.wikipedia.org/wiki/Elongated_triangular_bipyramid"]Elongated triangular bipyramid[/URL] (the Johnson solid J14) (f=9)
19. [URL="https://en.wikipedia.org/wiki/Octagonal_prism"]Octagonal prism[/URL] (the uniform prism with 8-gonal) (f=10)
20. [URL="https://en.wikipedia.org/wiki/Square_antiprism"]Square antiprism[/URL] (the uniform antiprism with 4-gonal) (f=10)
21. [URL="https://en.wikipedia.org/wiki/Square_cupola"]Square cupola[/URL] (the Johnson solid J4) (f=10)
22. [URL="https://en.wikipedia.org/wiki/Pentagonal_bipyramid"]Pentagonal bipyramid[/URL] (the Johnson solid J13) (f=10)
23. [URL="https://en.wikipedia.org/wiki/Augmented_pentagonal_prism"]Augmented pentagonal prism[/URL] (the Johnson solid J52) (f=10)
24. [URL="https://en.wikipedia.org/wiki/Augmented_tridiminished_icosahedron"]Augmented tridiminished icosahedron[/URL] (the Johnson solid J64) (f=10)
25. [URL="https://en.wikipedia.org/wiki/Enneagonal_prism"]Enneagonal prism[/URL] (the uniform prism with 9-gonal) (f=11)
26. [URL="https://en.wikipedia.org/wiki/Elongated_pentagonal_pyramid"]Elongated pentagonal pyramid[/URL] (the Johnson solid J9) (f=11)
27. [URL="https://en.wikipedia.org/wiki/Biaugmented_triangular_prism"]Biaugmented triangular prism[/URL] (the Johnson solid J50) (f=11)
28. [URL="https://en.wikipedia.org/wiki/Augmented_hexagonal_prism"]Augmented hexagonal prism[/URL] (the Johnson solid J54) (f=11)
29. [URL="https://en.wikipedia.org/wiki/Regular_dodecahedron"]Regular dodecahedron[/URL] (the Platonic solid with 12 faces) (f=12)
30. [URL="https://en.wikipedia.org/wiki/Decagonal_prism"]Decagonal prism[/URL] (the uniform prism with 10-gonal) (f=12)
31. [URL="https://en.wikipedia.org/wiki/Pentagonal_antiprism"]Pentagonal antiprism[/URL] (the uniform antiprism with 5-gonal) (f=12)
32. [URL="https://en.wikipedia.org/wiki/Pentagonal_cupola"]Pentagonal cupola[/URL] (the Johnson solid J5) (f=12)
33. [URL="https://en.wikipedia.org/wiki/Elongated_square_bipyramid"]Elongated square bipyramid[/URL] (the Johnson solid J15) (f=12)
34. [URL="https://en.wikipedia.org/wiki/Metabidiminished_icosahedron"]Metabidiminished icosahedron[/URL] (the Johnson solid J62) (f=12)
35. [URL="https://en.wikipedia.org/wiki/Snub_disphenoid"]Snub disphenoid[/URL] (the Johnson solid J84) (f=12)
36. [URL="https://en.wikipedia.org/wiki/Hendecagonal_prism"]Hendecagonal prism[/URL] (the uniform prism with 11-gonal) (f=13)
37. [URL="https://en.wikipedia.org/wiki/Gyroelongated_square_pyramid"]Gyroelongated square pyramid[/URL] (the Johnson solid J10) (f=13)
38. [URL="https://en.wikipedia.org/wiki/Biaugmented_pentagonal_prism"]Biaugmented pentagonal prism[/URL] (the Johnson solid J53) (f=13)
39. [URL="https://en.wikipedia.org/wiki/Dodecagonal_prism"]Dodecagonal prism[/URL] (the uniform prism with 12-gonal) (f=14)
40. [URL="https://en.wikipedia.org/wiki/Hexagonal_antiprism"]Hexagonal antiprism[/URL] (the uniform antiprism with 6-gonal) (f=14)
41. [URL="https://en.wikipedia.org/wiki/Cuboctahedron"]Cuboctahedron[/URL] (an Archimedean solid) (f=14)
42. [URL="https://en.wikipedia.org/wiki/Truncated_cube"]Truncated cube[/URL] (an Archimedean solid) (f=14)
43. [URL="https://en.wikipedia.org/wiki/Truncated_octahedron"]Truncated octahedron[/URL] (an Archimedean solid) (f=14)
44. [URL="https://en.wikipedia.org/wiki/Elongated_triangular_cupola"]Elongated triangular cupola[/URL] (the Johnson solid J18) (f=14)
45. [URL="https://en.wikipedia.org/wiki/Triangular_orthobicupola"]Triangular orthobicupola[/URL] (the Johnson solid J27) (f=14)
46. [URL="https://en.wikipedia.org/wiki/Triaugmented_triangular_prism"]Triaugmented triangular prism[/URL] (the Johnson solid J51) (f=14)
47. [URL="https://en.wikipedia.org/wiki/Parabiaugmented_hexagonal_prism"]Parabiaugmented hexagonal prism[/URL] (the Johnson solid J55) (f=14)
48. [URL="https://en.wikipedia.org/wiki/Metabiaugmented_hexagonal_prism"]Metabiaugmented hexagonal prism[/URL] (the Johnson solid J56) (f=14)
49. [URL="https://en.wikipedia.org/wiki/Augmented_truncated_tetrahedron"]Augmented truncated tetrahedron[/URL] (the Johnson solid J65) (f=14)
50. [URL="https://en.wikipedia.org/wiki/Sphenocorona"]Sphenocorona[/URL] (the Johnson solid J86) (f=14)
51. [URL="https://en.wikipedia.org/wiki/Bilunabirotunda"]Bilunabirotunda[/URL] (the Johnson solid J91) (f=14)
...
187. [URL="https://en.wikipedia.org/wiki/Rhombicosidodecahedron"]Rhombicosidodecahedron[/URL] (an Archimedean solid) (f=62)
188. [URL="https://en.wikipedia.org/wiki/Truncated_icosidodecahedron"]Truncated icosidodecahedron[/URL] (an Archimedean solid) (f=62)
189. [URL="https://en.wikipedia.org/wiki/Triaugmented_truncated_dodecahedron"]Triaugmented truncated dodecahedron[/URL] (the Johnson solid J71) (f=62)
190. [URL="https://en.wikipedia.org/wiki/Gyrate_rhombicosidodecahedron"]Gyrate rhombicosidodecahedron[/URL] (the Johnson solid J72) (f=62)
191. [URL="https://en.wikipedia.org/wiki/Parabigyrate_rhombicosidodecahedron"]Parabigyrate rhombicosidodecahedron[/URL] (the Johnson solid J73) (f=62)
192. [URL="https://en.wikipedia.org/wiki/Metabigyrate_rhombicosidodecahedron"]Metabigyrate rhombicosidodecahedron[/URL] (the Johnson solid J74) (f=62)
193. [URL="https://en.wikipedia.org/wiki/Trigyrate_rhombicosidodecahedron"]Trigyrate rhombicosidodecahedron[/URL] (the Johnson solid J75) (f=62)
...
239. [URL="https://en.wikipedia.org/wiki/Snub_dodecahedron"]Snub dodecahedron[/URL] (an Archimedean solid) (f=92)
...

Since all such solids (i.e. solids with [URL="https://en.wikipedia.org/wiki/Regular_polygon"]regular polygon[/URL] faces) have edges lengths all equal, we can ask ....

* If the edges of these solids have length 1, then the volume of these solids is? Also the surface area of these solids?
* The [URL="https://en.wikipedia.org/wiki/Dihedral_angle"]dihedral angles[/URL] and [URL="https://en.wikipedia.org/wiki/Solid_angle"]solid angles[/URL] for the vertices of these solids?
* The [URL="https://en.wikipedia.org/wiki/Dual_polyhedron"]dual polyhedron[/URL] of these solids? (There seems to be no name for the dual polyhedron of Johnson solids)
* The [URL="https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol"]Schläfli symbol[/URL] and [URL="https://en.wikipedia.org/wiki/Coxeter%E2%80%93Dynkin_diagram"]Coxeter–Dynkin diagram[/URL] for these solids?
* The [URL="https://en.wikipedia.org/wiki/List_of_spherical_symmetry_groups"]symmetry group[/URL] of these solids?
* The [URL="https://en.wikipedia.org/wiki/Schlegel_diagram"]Schlegel diagram[/URL] of these solids and their dual polyhedrons, also the number of (undirected) [URL="https://en.wikipedia.org/wiki/Eulerian_path"]Euler path[/URL], Euler cycle, [URL="https://en.wikipedia.org/wiki/Hamiltonian_path"]Hamiltonian path[/URL], Hamiltonian cycle, of the Schlegel diagram of them?

There is an order of [URL="https://en.wikipedia.org/wiki/Finite_group"]finite groups[/URL]: by order, then by index from [URL="http://www.icm.tu-bs.de/ag_algebra/software/small/"]the small groups library[/URL], starting at 1

1. C1 (trivial group) (cyclic group with order 1) (ord=1)
2. C2 (cyclic group with order 2) (ord=2)
3. C3 (cyclic group with order 3) = A3 (Alternating group on 3 letters) (ord=3)
4. C4 (cyclic group with order 4) (ord=4)
5. C2xC2 (Klein 4-group) (ord=4)
6. C5 (cyclic group with order 5) (ord=5)
7. S3 (Symmetric group on 3 letters) (ord=6)
8. C6 (cyclic group with order 6) (ord=6)
9. C7 (cyclic group with order 7) (ord=7)
10. C8 (cyclic group with order 8) (ord=8)
11. C4xC2 (ord=8)
12. D4 (Dihedral group) (ord=8)
13. Q8 (Quaternion group) (ord=8)
14. C2xC2xC2 (ord=8)
15. C9 (cyclic group with order 9) (ord=9)
16. C3xC3 (ord=9)
17. D5 (Dihedral group) (ord=10)
18. C10 (cyclic group with order 10) (ord=10)
19. C11 (cyclic group with order 11) (ord=11)
20. Dic3 (Dicyclic group) (ord=12)
21. C12 (cyclic group with order 12) (ord=12)
22. A4 (Alternating group on 4 letters) (ord=12)
23. D6 (Dihedral group) (ord=12)
24. C6xC2 (ord=12)
25. C13 (cyclic group with order 13) (ord=13)
26. D7 (Dihedral group) (ord=14)
27. C14 (cyclic group with order 14) (ord=14)
28. C15 (cyclic group with order 15) (ord=15)
29. C16 (cyclic group with order 16) (ord=16)
30. C4xC4 (ord=16)
31. The semidirect product of C2xC2 and C4 (ord=16)
32. The semidirect product of C4 and C4 (ord=16)
33. C8xC2 (ord=16)
34. M4(2) (Modular maximal-cyclic group) (ord=16)
35. D8 (Dihedral group) (ord=16)
36. SD16 (Semidihedral group) (ord=16)
37. Q16 (Generalised quaternion group) (ord=16)
38. C4xC4xC2 (ord=16)
39. D4xC2 (ord=16)
40. Q8xC2 (ord=16)
41. Central product of C4 and D4 (ord=16)
42. C2xC2xC2xC2 (ord=16)
...

However, I do not think that the sort in the small groups library is reasonable, e.g. S3 (=D3) is before C6, D4 is after C8, D5 is before C10, and D6 is after C12, ..., I think that the cyclic group should be before all other groups with same order, and Abelian group should be before non-Abelian group with same order (Dn can be after the Abelian groups and before all other non-Abelian groups with order 2*n)