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

7,984 groups of the catalog are in this class; the 7,886 with an abelian direct factor are hidden.

grouporder× stem
D816×1◆ stem
SD1616×1◆ stem
Q1616×1◆ stem
#10232×2
#10332×2
#10432×2
#10632×2
#10732×2
#10832×2
#13532×2
D4⋊C864×4
Q8⋊C864×4
#33464×4
#33564×4
#33964×4
#34064×4
#34164×4
#35064×4
#36464×4
#41664×4
#42764×4
#44364×4
#1276128×8
#1277128×8
#1278128×8
#1279128×8
D4⋊C16128×8
Q8⋊C16128×8
#1367128×8
#1370128×8
#1376128×8
#1379128×8
#1401128×8
#1422128×8
#1476128×8
#1564128×8
#1735128×8
#1764128×8
#1772128×8
#2170128×8
#7015256×16
#7138256×16
#7141256×16
#7148256×16
#7151256×16
#7163256×16
#7167256×16
D4⋊C32256×16
Q8⋊C32256×16
#7455256×16
#7458256×16
#7467256×16
#7470256×16
#7514256×16
#7544256×16
#7872256×16
#7890256×16
#8184256×16
#8266256×16
#10706256×16
#11613256×16
#12012256×16
#12614256×16
#12664256×16
#13647256×16
#92897512×32
#92902512×32
#93257512×32
#93260512×32
#93991512×32
#93994512×32
#94010512×32
#94015512×32
#94032512×32
#94036512×32
D4⋊C64512×32
Q8⋊C64512×32
#94745512×32
#94747512×32
#94757512×32
#94760512×32
#94818512×32
#94839512×32
#94879512×32
#105277512×32
#105360512×32
#126668512×32
#126794512×32
#128249512×32
#128620512×32
#129043512×32
#129075512×32
#146318512×32
#150083512×32
#151041512×32
#151900512×32
#151986512×32
#153586512×32

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.