Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#11724633 |
| Fitting | C24×C33 ⋊ C22 |
| Sylow | C33 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#1113 |
| commutator, cosocle | C2×C33 | C25 |
| Frattini | C2 | C23×#1113 |
| Fitting | C24×C33 | C22 |
| socle | C22×C33 | C24 |
| hypercenter | C24 | #1113 |
| hypocenter | C33 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C24 | |
4434 conjugacy classes of subgroups; only the 663 normal subgroups are shown.
This group has 4434 conjugacy classes of subgroups — too many to draw. Only its 663 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47758);