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 ◆.

725 groups of the catalog are in this class; the 660 with an abelian direct factor are hidden.

grouporder× stem
AΣL(1,𝔽16)64×1◆ stem
#35264×1◆ stem
#1316128×2
#1317128×2
#1320128×2
#1326128×2
#1328128×2
#1390128×2
#1391128×2
#1392128×2
#2120128×2
#7054256×4
#7055256×4
#7064256×4
#7187256×4
#7188256×4
#7191256×4
#7192256×4
#7203256×4
#7204256×4
#7207256×4
#7336256×4
#7365256×4
#11425256×4
#11452256×4
#11471256×4
#11472256×4
#11483256×4
#12573256×4
#12623256×4
#92963512×8
#92964512×8
#92965512×8
#92966512×8
#92995512×8
#92996512×8
#93282512×8
#93284512×8
#93286512×8
#93669512×8
#93681512×8
#93730512×8
#94064512×8
#94066512×8
#94126512×8
#94512512×8
#94578512×8
#122224512×8
#122313512×8
#122358512×8
#122359512×8
#126908512×8
#126947512×8
#126949512×8
#126957512×8
#127066512×8
#127070512×8
#128308512×8
#128318512×8
#128344512×8
#128352512×8
#128461512×8
#146352512×8
#149669512×8
#480090512×8

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.