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 ◆.

7,106 groups of the catalog are in this class; the 6,703 with an abelian direct factor are hidden.

grouporder× stem
#12032×1◆ stem
#12132×1◆ stem
C22⋊Q832×1◆ stem
#12332×1◆ stem
#12432×1◆ stem
#12532×1◆ stem
#12632×1◆ stem
#12732×1◆ stem
C4⋊Q832×1◆ stem
#37964×2
#38064×2
#38164×2
#38264×2
#38364×2
#38464×2
#38564×2
#38664×2
#38764×2
#38864×2
#38964×2
#39064×2
#39164×2
#40764×2
#40864×2
#42364×2
#42464×2
#43264×2
#43364×2
#43564×2
#43664×2
#44664×2
#52564×2
#52964×2
#53264×2
#53364×2
C22⋊#95128×4
#1434128×4
C4⋊#95128×4
#1436128×4
#1437128×4
#1451128×4
#1452128×4
#1453128×4
#1454128×4
#1455128×4
#1728128×4
#1729128×4
#1745128×4
#1746128×4
#1749128×4
#1750128×4
#1752128×4
#1753128×4
#1767128×4
#1768128×4
#1770128×4
#1771128×4
#1779128×4
#1797128×4
#1817128×4
#1831128×4
#1839128×4
#1842128×4
#1850128×4
#1869128×4
#1916128×4
#1923128×4
#1939128×4
#1954128×4
#1984128×4
#2112128×4
#2113128×4
#2150128×4
#2151128×4
#2163128×4
#2164128×4
#2168128×4
#2169128×4
#2183128×4
#2280128×4
#2283128×4
#2285128×4
#2290128×4
#2293128×4
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#2920128×4
#2922128×4
#2929128×4
#2931128×4
#2960128×4
#2965128×4
#2966128×4
#3444128×4
#7554256×8
#7842256×8
#7843256×8
#7844256×8
#7845256×8
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#8132256×8
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#8260256×8
#8261256×8
#8262256×8
#8264256×8
#10693256×8
#10694256×8
#10696256×8
#10712256×8
#10714256×8
#10795256×8
#10797256×8
#10832256×8
#10834256×8
#10874256×8
#10876256×8
#10921256×8
#11401256×8
#11402256×8
#11408256×8
#11409256×8
#11411256×8
#11412256×8
#12544256×8
#12545256×8
#12591256×8
#12592256×8
#12599256×8
#12600256×8
#12611256×8
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#12653256×8
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#12679256×8
#12783256×8
#12844256×8
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#12894256×8
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#12912256×8
#12976256×8
#13104256×8
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#13159256×8
#13189256×8
#13263256×8
#13553256×8
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#13627256×8
#13628256×8
#13640256×8
#13641256×8
#13645256×8
#13646256×8
#13660256×8
#13753256×8
#13755256×8
#13756256×8
#13757256×8
#13759256×8
#13761256×8
#17323256×8
#17328256×8
#17335256×8
#20364256×8
#20366256×8
#20383256×8
#20385256×8
#20435256×8
#20437256×8
#20462256×8
#20463256×8
#20472256×8
#20473256×8
#20474256×8
#20476256×8
#20484256×8
#20485256×8
#20516256×8
#20714256×8
#20777256×8
#20808256×8
#21101256×8
#21264256×8
#21339256×8
#21349256×8
#21373256×8
#21430256×8
#33418256×8
#33420256×8
#33432256×8
#33434256×8
#33474256×8
#33479256×8
#33480256×8
#34012256×8
#34026256×8
#60204256×8
#94853512×16
#94854512×16
#94855512×16
#94856512×16
#94857512×16
#105246512×16
#105247512×16
#105338512×16
#105339512×16
#105340512×16
#105342512×16
C22⋊#1273512×16
#108386512×16
C4⋊#1273512×16
#109460512×16
#110042512×16
#122207512×16
#122208512×16
#122209512×16
#122210512×16
#122211512×16
#123422512×16
#123424512×16
#126521512×16
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#126526512×16
#126527512×16
#126531512×16
#126532512×16
#126547512×16
#126548512×16
#126560512×16
#126572512×16
#128204512×16
#128205512×16
#128209512×16
#128214512×16
#128240512×16
#128241512×16
#128246512×16
#128247512×16
#128251512×16
#128263512×16
#128271512×16
#128282512×16
#129036512×16
#129037512×16
#129041512×16
#129042512×16
#129045512×16
#129057512×16
#129065512×16
#129070512×16
#146295512×16
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#146316512×16
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#146808512×16
#146832512×16
#146942512×16
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#152790512×16
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#152845512×16
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#153427512×16
#153428512×16
#153567512×16
#153568512×16
#153580512×16
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#153600512×16
#153711512×16
#180800512×16
#180806512×16
#180813512×16
#180815512×16
#180825512×16
#180839512×16
#181847512×16
#181849512×16
#181885512×16
#181887512×16
#183281512×16
#183283512×16
#183722512×16
#183723512×16
#183724512×16
#183726512×16
#183731512×16
#183740512×16
#183838512×16
#184049512×16
#184378512×16
#353118512×16
#353159512×16
#353160512×16
#353167512×16
#353168512×16
#353196512×16
#353201512×16
#353231512×16
#353236512×16
#353563512×16
#353571512×16
#353716512×16
#353724512×16
#353758512×16
#353765512×16
#353835512×16
#353839512×16
#353986512×16
#354006512×16
#354126512×16
#354134512×16
#479914512×16
#479916512×16
#479924512×16
#479926512×16
#479941512×16
#479942512×16
#492920512×16
#492921512×16
#492964512×16
#492966512×16
#493116512×16
#493118512×16
#493164512×16
#493165512×16
#493188512×16
#493189512×16
#493190512×16
#493192512×16
#493205512×16
#493218512×16
#493258512×16
#493928512×16
#494105512×16
#494170512×16
#494937512×16
#495274512×16
#495433512×16
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#512728512×16
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#512742512×16
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#512790512×16
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#513341512×16
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#6342481512×16
#6369980512×16
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#6370142512×16
#6371219512×16
#6372545512×16
#7622657512×16

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.