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

95 groups of the catalog are in this class; the 80 with an abelian direct factor are hidden.

grouporder× stem
((C3⋊D3)⋊C3)⋊C2108×1◆ stem
((C3⋊D3)⋊C3)⋊C4216×2
#6199216×2
((C3⋊D3)⋊C3)⋊C8432×4
#88434432×4
#88447432×4
((C3⋊D3)⋊C3)⋊C16864×8
#11720251864×8
#11720302864×8
#11720310864×8
((C3⋊D3)⋊C3)⋊C321,728×16
#131349141,728×16
#131349601,728×16
#131350621,728×16
#131350781,728×16

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.