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

3,210 groups of the catalog are in this class; the 3,145 with an abelian direct factor are hidden.

grouporder× stem
D1632×1◆ stem
SD3232×1◆ stem
Q3232×1◆ stem
#35764×2
#35864×2
#35964×2
#36664×2
#36764×2
#36864×2
#50864×2
#1331128×4
#1332128×4
#1335128×4
#1371128×4
#1372128×4
#1373128×4
#1380128×4
#1381128×4
#1382128×4
#2138128×4
#2158128×4
#2178128×4
#7068256×8
#7069256×8
#7100256×8
#7101256×8
#7154256×8
#7155256×8
#7160256×8
#7171256×8
#7172256×8
#7179256×8
#7381256×8
#7459256×8
#11591256×8
#12008256×8
#12046256×8
#12556256×8
#12641256×8
#12670256×8
#92911512×16
#92912512×16
#92931512×16
#92932512×16
#93264512×16
#93268512×16
#93271512×16
#93745512×16
#93753512×16
#93879512×16
#93883512×16
#94018512×16
#94040512×16
#94605512×16
#94748512×16
#122440512×16
#122930512×16
#126648512×16
#126769512×16
#128533512×16
#128673512×16
#129095512×16
#129142512×16
#146331512×16
#151072512×16

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.