Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#11722850 |
| Sylow | C33 ⋊ C2×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D3×#625 |
| commutator, cosocle | C33×C4 | C24 |
| Frattini | C4 | C2×D33 |
| Fitting | C2×C33×C4 | C23 |
| socle | C22×C33 | C2×D4 |
| hypercenter | C2×C4 | D33 |
| hypocenter | C33 | C2×#135 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
1298 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 1298 conjugacy classes of subgroups — too many to draw. Only its 212 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 36669);