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 ◆.

1,433 groups of the catalog are in this class; the 1,356 with an abelian direct factor are hidden.

grouporder× stem
H(3,ℤ/4)64×1◆ stem
#33864×1◆ stem
#34764×1◆ stem
#1274128×2
#1275128×2
#1313128×2
#1375128×2
#1398128×2
#1747128×2
#2110128×2
#7014256×4
#7053256×4
#7137256×4
#7142256×4
#7145256×4
#7330256×4
#7332256×4
#7466256×4
#7511256×4
#7854256×4
#7860256×4
#8137256×4
#8138256×4
#10699256×4
#11404256×4
#11405256×4
#11406256×4
#12541256×4
#12542256×4
#12547256×4
#12593256×4
#12594256×4
#13551256×4
#92896512×8
#93249512×8
#93666512×8
#93667512×8
#93990512×8
#93998512×8
#94005512×8
#94505512×8
#94508512×8
#94756512×8
#94815512×8
#94863512×8
#96658512×8
#96659512×8
#105257512×8
#122213512×8
#122214512×8
#126524512×8
#126529512×8
#126533512×8
#126550512×8
#126551512×8
#126592512×8
#126612512×8
#128206512×8
#128207512×8
#128208512×8
#128225512×8
#128228512×8
#146298512×8
#146317512×8
#149495512×8
#149496512×8
#149501512×8
#149502512×8
#151750512×8
#151751512×8
#151756512×8
#151879512×8
#151880512×8
#153425512×8
#181820512×8
#479910512×8
#492923512×8

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.