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 8: every member's order is a multiple of it, and the members attaining it — the stem groups — are marked ◆.

10,984 groups of the catalog are in this class; the 10,906 with an abelian direct factor are hidden.

grouporder× stem
D48×1◆ stem
Q88×1◆ stem
C22⋊C416×2
C4⋊C416×2
M1616×2
#4116×2
#9532×4
#9732×4
C22⋊C832×4
C4⋊C832×4
M3232×4
#11732×4
#13132×4
#32264×8
#33664×8
#34664×8
C22⋊C1664×8
C4⋊C1664×8
M6464×8
#37664×8
#40564×8
#43164×8
#50464×8
#1273128×16
#1311128×16
#1312128×16
#1374128×16
#1397128×16
C22⋊C32128×16
C4⋊C32128×16
M128128×16
#1450128×16
#1726128×16
#1744128×16
#2108128×16
#2162128×16
#2258128×16
#7052256×32
#7136256×32
#7329256×32
#7331256×32
#7465256×32
#7510256×32
C22⋊C64256×32
C4⋊C64256×32
M256256×32
#7841256×32
#8131256×32
#11399256×32
#11410256×32
#12539256×32
#12590256×32
#13549256×32
#13639256×32
#13737256×32
#92895512×64
#93664512×64
#93665512×64
#93989512×64
#94504512×64
#94507512×64
#94755512×64
#94814512×64
C22⋊C128512×64
C4⋊C128512×64
M512512×64
#94852512×64
#122206512×64
#126519512×64
#126530512×64
#128203512×64
#149493512×64
#149510512×64
#151748512×64
#151876512×64
#153423512×64
#153579512×64
#153701512×64

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.