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 ◆.

824 groups of the catalog are in this class; the 757 with an abelian direct factor are hidden.

grouporder× stem
#36064×1◆ stem
#36164×1◆ stem
#36264×1◆ stem
#1333128×2
#1334128×2
#1336128×2
#1386128×2
#1387128×2
#1388128×2
#1389128×2
#2145128×2
#2148128×2
#7070256×4
#7071256×4
#7158256×4
#7159256×4
#7162256×4
#7175256×4
#7176256×4
#7177256×4
#7178256×4
#7181256×4
#7182256×4
#7382256×4
#11597256×4
#11598256×4
#11599256×4
#11600256×4
#11604256×4
#12567256×4
#12569256×4
#12646256×4
#92919512×8
#92920512×8
#92939512×8
#92940512×8
#92941512×8
#92942512×8
#93266512×8
#93267512×8
#93270512×8
#93273512×8
#93755512×8
#93760512×8
#94020512×8
#94042512×8
#94043512×8
#94606512×8
#122464512×8
#122465512×8
#122466512×8
#122467512×8
#126664512×8
#126666512×8
#126779512×8
#126781512×8
#126783512×8
#126785512×8
#126786512×8
#126787512×8
#126793512×8
#128528512×8
#128537512×8
#128675512×8
#146343512×8
#150053512×8
#481065512×8

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.