The Fischer groups are named by analogy with the large Mathieu groups. In Fi22 a maximal set of 3-transpositions all commuting with one another has size 22 and is called a "basic" set. There are 1024 3-transpositions, called "anabasic" that do not commute with any in the particular basic set. Any one of other 2364, called "hexadic", commutes with 6 basic ones. The sets of 6 form an S(3,6,22) Steiner system, whose symmetry group is M22. A basic set generates an abelian group of order 210, which extends in Fi22 to a subgroup 210:M22.