Finite Group Databasebeta

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Isoclinism class

Groups whose commutator structure agrees: there are isomorphisms G/Z(G) → H/Z(H) and G′ → H′ carrying one commutator map to the other. The class spans orders — G × A is isoclinic to G for every abelian A.

Stem order 27: every member's order is a multiple of it, and the members attaining it — the stem groups — are marked ◆.

1,427 groups of the catalog are in this class; the 1,390 with an abelian direct factor are hidden.

The class itself is infinite — G × A is isoclinic to G for every abelian A, and there are abelian groups of every order — so a table shows the part of it the catalog names, never the whole.