Finite Group Databasebeta

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

1,502 groups of the catalog are in this class; the 1,338 with an abelian direct factor are hidden.

grouporder× stem
#57664×1◆ stem
#57764×1◆ stem
#57864×1◆ stem
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#3584128×2
#17438256×4
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#180896512×8
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#7622587512×8

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.