Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#11723728 |
| Sylow | C33 ⋊ C2×#127 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×#1113 |
| commutator, cosocle | C22×C33 | C24 |
| Frattini | C22 | C22×#1113 |
| Fitting | C24×C33 | C22 |
| socle | C23×C33 | C23 |
| hypercenter | C24 | #1113 |
| hypocenter | C33 | C2×#127 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C23 ◁ C24 | |
2834 conjugacy classes of subgroups; only the 295 normal subgroups are shown.
This group has 2834 conjugacy classes of subgroups — too many to draw. Only its 295 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 44863);