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 ◆.

747 groups of the catalog are in this class; the 676 with an abelian direct factor are hidden.

grouporder× stem
B(4,𝔽2)64×1◆ stem
#45864×1◆ stem
#1790128×2
#1791128×2
#1867128×2
#1868128×2
#1889128×2
#1890128×2
#1891128×2
#3025128×2
#8316256×4
#8851256×4
#8965256×4
#8966256×4
#10745256×4
#10746256×4
#10747256×4
#10748256×4
#10918256×4
#10919256×4
#10920256×4
#10935256×4
#10936256×4
#10937256×4
#12684256×4
#12915256×4
#12981256×4
#20608256×4
#20888256×4
#20959256×4
#21483256×4
#106314512×8
#106316512×8
#106318512×8
#106320512×8
#106322512×8
#106324512×8
#109580512×8
#109582512×8
#109584512×8
#109586512×8
#110044512×8
#110045512×8
#110048512×8
#123463512×8
#123464512×8
#123465512×8
#123466512×8
#129567512×8
#135934512×8
#136807512×8
#136835512×8
#146374512×8
#146400512×8
#146408512×8
#147117512×8
#147140512×8
#147360512×8
#152006512×8
#152413512×8
#152491512×8
#184904512×8
#192027512×8
#193864512×8
#353426512×8
#353979512×8
#354030512×8
#354316512×8
#354477512×8
#354787512×8
#495677512×8

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.