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 ◆.

2,142 groups of the catalog are in this class; the 2,039 with an abelian direct factor are hidden.

grouporder× stem
#9932×1◆ stem
#10032×1◆ stem
#10132×1◆ stem
#32364×2
#32464×2
#34264×2
#34364×2
#34464×2
#34964×2
#41064×2
#41364×2
#1280128×4
#1281128×4
#1282128×4
#1283128×4
#1314128×4
#1322128×4
#1377128×4
#1378128×4
#1400128×4
#1458128×4
#1459128×4
#1460128×4
#1461128×4
#1740128×4
#1742128×4
#1757128×4
#2115128×4
#7016256×8
#7139256×8
#7140256×8
#7183256×8
#7195256×8
#7199256×8
#7209256×8
#7334256×8
#7362256×8
#7468256×8
#7469256×8
#7513256×8
#7906256×8
#7907256×8
#7908256×8
#7918256×8
#7924256×8
#7932256×8
#8156256×8
#10711256×8
#11459256×8
#11465256×8
#11487256×8
#11493256×8
#12551256×8
#12602256×8
#12613256×8
#13556256×8
#17349256×8
#92898512×16
#92901512×16
#93274512×16
#93278512×16
#93992512×16
#93993512×16
#94060512×16
#94116512×16
#94122512×16
#94164512×16
#94510512×16
#94575512×16
#94758512×16
#94759512×16
#94817512×16
#94915512×16
#94916512×16
#94917512×16
#105300512×16
#105382512×16
#126555512×16
#126557512×16
#126927512×16
#126937512×16
#126961512×16
#127037512×16
#127085512×16
#127167512×16
#128248512×16
#128295512×16
#128327512×16
#128428512×16
#128431512×16
#146302512×16
#146320512×16
#149653512×16
#149663512×16
#149725512×16
#149733512×16
#151760512×16
#151888512×16
#151899512×16
#153430512×16
#180998512×16
#182035512×16
#480176512×16

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.