Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C23×C3 |
| direct product | C9×#5950 |
| Fitting | C9×#585 ⋊ C3 |
| Sylow | #585 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C9 | C22⋊A4 |
| commutator, hypocenter | H(5,𝔽2) | C2×C3×C9 |
| Frattini, socle | C2×C3 | C2×C3×C22⋊A4 |
| Fitting | C9×#585 | C3 |
| prime radical | C3×#5950 | C3 |
| cosocle | C3×H(5,𝔽2) | C2×C32 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C4×C9 | |
567 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 567 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
198 conjugacy classes — view the full character table.
| maximal subgroup of | 3−1+2×#5950 |
| maximal normal subgroup of | 3−1+2×#5950 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30937);