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 ◆.

2,308 groups of the catalog are in this class; the 2,151 with an abelian direct factor are hidden.

grouporder× stem
AGL(1,ℤ/8)32×1◆ stem
#13732×1◆ stem
#41764×2
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#41964×2
#42164×2
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#43064×2
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#44264×2
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#57564×2
#1477128×4
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#1565128×4
SD16⋊C8128×4
#1579128×4
#1580128×4
#1733128×4
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#33441256×8
#94880512×16
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#150091512×16
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#483604512×16
#492973512×16
#493123512×16
#493213512×16
#512749512×16

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.