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

127 groups of the catalog are in this class; the 112 with an abelian direct factor are hidden.

group	order	× stem
GO⁺(4,𝔽₂)	72	×1	◆ stem
D₃²⋊C₄	144	×2
#3773	144	×2
D₃²⋊C₈	288	×4
#63732	288	×4
#63737	288	×4
D₃²⋊C₁₆	576	×8
#10590022	576	×8
#10590043	576	×8
#10590044	576	×8
#11770702	1,152	×16
#11772270	1,152	×16
#11772271	1,152	×16
D₃²⋊C₃₂	1,152	×16
#11773017	1,152	×16

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.