In mathematics, a fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by John Horton Conway, Olaf Delgado Friedrichs, and Daniel H. Huson et al. (2001), who introduced a system of notation for 3-dimensional fibrifolds and used this to assign names to the 219 affine space group types. 184 of these are considered reducible, and 35 irreducible.
Irreducible cubic space groups
The 35/36 irreducible cubic space groups in fibrifold and international index and Hermann–Mauguin notation. 212 and 213 are enantiomorphous pairs giving the same fibrifold notation.
The 35 irreducible space groups correspond to the cubic space group.
35 irreducible space groups
8o:2
4−:2
4o:2
4+:2
2−:2
2o:2
2+:2
1o:2
8o
4−
4o
4+
2−
2o
2+
1o
8o/4
4−/4
4o/4
4+/4
2−/4
2o/4
2+/4
1o/4
8−o
8oo
8+o
4− −
4−o
4oo
4+o
4++
2−o
2oo
2+o
36 cubic groups
Class Point group
Hexoctahedral *432 (m3m)
Hextetrahedral *332 (43m)
Gyroidal 432 (432)
Diploidal 3*2 (m3)
Tetartoidal 332 (23)
bc lattice (I)
8o:2 (Im3m)
4o:2 (I43m)
8+o (I432)
8−o (I3)
4oo (I23)
nc lattice (P)
4−:2 (Pm3m)
2o:2 (P43m)
4−o (P432)
4− (Pm3)
2o (P23)
4+:2 (Pn3m)
4+ (P4232)
4+o (Pn3)
fc lattice (F)
2−:2 (Fm3m)
1o:2 (F43m)
2−o (F432)
2− (Fm3)
1o (F23)
2+:2 (Fd3m)
2+ (F4132)
2+o (Fd3)
Other lattice groups
8o (Pm3n) 8oo (Pn3n) 4− − (Fm3c) 4++ (Fd3c)
4o (P43n) 2oo (F43c)
Achiral quarter groups
8o/4 (Ia3d)
4o/4 (I43d)
4+/4 (I4132) 2+/4 (P4332, P4132)
2−/4 (Pa3) 4−/4 (Ia3)
1o/4 (P213) 2o/4 (I213)
8 primary hexoctahedral hextetrahedral lattices of the cubic space groups
The fibrifold cubic subgroup structure shown is based on extending symmetry of the tetragonal disphenoid fundamental domain of space group 216, similar to the square
