Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #10595527⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #10595527⋊C3 |
| Fitting, derived | C24×C32 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C24×C32 | C22×C3 |
| Frattini | C22 | D32×A4 |
| socle | C22×C32 | C22×A4 |
| LCS | G ▷ C24×C32 | |
| UCS | C1 | |
762 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 762 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^2,3,2^4,3^2|d^a=df,e^a=efg,h^a=h^2,d^b=df,e^b=efg,i^b=i^2,d^c=de,e^c=d,f^c=g,g^c=fg48 conjugacy classes — view the full character table.
| maximal subgroup of | #72210357, C3×#10595527⋊C3, #72220666, … |
| maximal normal subgroup of | #72210357, C3×#10595527⋊C3, D32×#120⋊C3, … |
| maximal quotient of | C3×#10595527⋊C3, #72220666, #72220699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32233);