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 ◆.

151 groups of the catalog are in this class; the 137 with an abelian direct factor are hidden.

group	order	× stem
AGL(1,ℤ/9)	54	×1	◆ stem
3₋¹⁺²⋊C₄	108	×2
D₉⋊C₉	162	×3
3₋¹⁺²⋊C₈	216	×4
Dic₉⋊C₉	324	×6
3₋¹⁺²⋊C₁₆	432	×8
D₉⋊C₂₇	486	×9
(C₉⋊C₈)⋊C₉	648	×12
3₋¹⁺²⋊C₃₂	864	×16
Dic₉⋊C₂₇	972	×18
(C₉⋊C₉)⋊C₁₆	1,296	×24
D₉⋊C₈₁	1,458	×27
3₋¹⁺²⋊C₆₄	1,728	×32
(C₉⋊C₈)⋊C₂₇	1,944	×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.