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 ◆.

832 groups of the catalog are in this class; the 787 with an abelian direct factor are hidden.

grouporder× stem
#36564×1◆ stem
#1368128×2
#1369128×2
#1383128×2
#1384128×2
#1385128×2
#2155128×2
#7099256×4
#7156256×4
#7157256×4
#7161256×4
#7173256×4
#7174256×4
#7180256×4
#7456256×4
#11999256×4
#12000256×4
#12002256×4
#12563256×4
#12668256×4
#92913512×8
#92914512×8
#92933512×8
#92934512×8
#93265512×8
#93269512×8
#93272512×8
#93876512×8
#94019512×8
#94041512×8
#94746512×8
#122919512×8
#122920512×8
#126653512×8
#126654512×8
#126656512×8
#126660512×8
#126775512×8
#126777512×8
#129088512×8
#129090512×8
#129137512×8
#146335512×8
#151022512×8
#483322512×8

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.