Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×B(4,𝔽2)⋊C9 |
| unambiguous semidirect product | (C3×B(4,𝔽2))⋊C9 |
| derived | H(5,𝔽2) ⋊ C2×C3×C9 |
| Sylow | B(4,𝔽2) ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C32 | #120⋊C3 |
| commutator, hypocenter | H(5,𝔽2) | C2×C3×C9 |
| Frattini | C23×C3 | C2×C3×A4 |
| Fitting, prime radical | C32×B(4,𝔽2) | C3 |
| cosocle | C3×H(5,𝔽2) | C2×C32 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C32 | |
318 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7319);