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 ◆.

1,364 groups of the catalog are in this class; the 1,220 with an abelian direct factor are hidden.

grouporder× stem
#56064×1◆ stem
H(3,𝔽4)64×1◆ stem
#56264×1◆ stem
#56364×1◆ stem
#56464×1◆ stem
#2375128×2
#2376128×2
#2377128×2
#2378128×2
#2379128×2
#2380128×2
#2381128×2
#2382128×2
#2383128×2
#2978128×2
#2980128×2
#2987128×2
#2995128×2
#3529128×2
#3531128×2
#13827256×4
#13828256×4
#13829256×4
#13830256×4
#13831256×4
#13832256×4
#13833256×4
#13834256×4
#17338256×4
#17339256×4
#17342256×4
#20529256×4
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#20539256×4
#20542256×4
#20551256×4
#20554256×4
#20563256×4
#20564256×4
#20576256×4
#23306256×4
#23314256×4
#23326256×4
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#23346256×4
#23355256×4
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#23369256×4
#23375256×4
#33492256×4
#33494256×4
#33501256×4
#33509256×4
#34342256×4
#34346256×4
#34348256×4
#34350256×4
#60281256×4
#60307256×4
#60332256×4
#62694256×4
#153730512×8
#153731512×8
#180830512×8
#180831512×8
#180835512×8
#180842512×8
#180845512×8
#184431512×8
#184434512×8
#184474512×8
#184482512×8
#184491512×8
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#184585512×8
#184611512×8
#184695512×8
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#353283512×8
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#498723512×8
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#498758512×8
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#512802512×8
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#512811512×8
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#6342505512×8
#6370243512×8
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#6407552512×8
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#6407752512×8
#7622734512×8
#7622759512×8
#7622784512×8
#7625931512×8
#10574594512×8

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.