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 ◆.

1,093 groups of the catalog are in this class; the 1,016 with an abelian direct factor are hidden.

grouporder× stem
#35364×1◆ stem
#35464×1◆ stem
#35564×1◆ stem
#35664×1◆ stem
#1318128×2
#1319128×2
#1321128×2
#1324128×2
#1325128×2
#1327128×2
#1393128×2
#1394128×2
#1395128×2
#2126128×2
#2132128×2
#7056256×4
#7057256×4
#7062256×4
#7063256×4
#7189256×4
#7190256×4
#7193256×4
#7194256×4
#7205256×4
#7206256×4
#7208256×4
#7337256×4
#7364256×4
#11431256×4
#11447256×4
#11475256×4
#11476256×4
#11477256×4
#11478256×4
#11485256×4
#12581256×4
#12585256×4
#12633256×4
#92967512×8
#92968512×8
#92969512×8
#92970512×8
#92997512×8
#92998512×8
#93283512×8
#93285512×8
#93287512×8
#93687512×8
#93695512×8
#93717512×8
#93723512×8
#94065512×8
#94067512×8
#94127512×8
#94513512×8
#94577512×8
#122242512×8
#122296512×8
#122374512×8
#122375512×8
#122376512×8
#122377512×8
#126914512×8
#126951512×8
#126952512×8
#126955512×8
#126959512×8
#127074512×8
#127078512×8
#128314512×8
#128322512×8
#128340512×8
#128348512×8
#128463512×8
#146358512×8
#149671512×8
#480104512×8

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.