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

154 groups of the catalog are in this class; the 140 with an abelian direct factor are hidden.

grouporder× stem
(C3⋊D3)⋊C354×1◆ stem
(C3⋊Dic3)⋊C3108×2
(C3⋊D3)⋊C9162×3
(C3⋊(C3⋊C8))⋊C3216×4
(C3⋊Dic3)⋊C9324×6
(C3⋊(C3⋊C16))⋊C3432×8
(C3⋊D3)⋊C27486×9
(C3⋊(C3⋊C8))⋊C9648×12
(C3⋊(C3⋊C32))⋊C3864×16
(C3⋊Dic3)⋊C27972×18
(C3⋊(C3⋊C16))⋊C91,296×24
(C3⋊D3)⋊C811,458×27
(C3⋊(C3⋊C64))⋊C31,728×32
(C3⋊(C3⋊C8))⋊C271,944×36

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.