Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C33 |
| direct product | H(5,𝔽2)×C3⋊(C3⋊D3) |
| Fitting | C33×H(5,𝔽2) ⋊ C2 |
| Sylow | C33 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C33 | C25 |
| Fitting | C33×H(5,𝔽2) | C2 |
| hypercenter | H(5,𝔽2) | C3⋊(C3⋊D3) |
| hypocenter | C33 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
12572 conjugacy classes of subgroups; only the 2213 normal subgroups are shown.
This group has 12572 conjugacy classes of subgroups — too many to draw. Only its 2213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
255 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47827);