Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D4×#157⋊D3 |
| Fitting | C33×D4 ⋊ D4 |
| Sylow | C33 ⋊ D42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#157⋊D3 |
| commutator, cosocle | C2×C3×C3⋊D3 | C24 |
| Fitting | C33×D4 | D4 |
| socle | C2×C33 | C22×D4 |
| hypercenter | D4 | #157⋊D3 |
| hypocenter | C33 | D42 |
| LCS | G ▷ C2×C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ D4 | |
1436 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 1436 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
75 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47215);