Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#5933 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | #1020 |
| commutator, hypocenter | #997 | C2×C9 |
| Frattini, socle | C2×C3 | C3×#1020 |
| Fitting | C9×H(5,𝔽2) | D3 |
| prime radical, cosocle | C3×#997 | C2×C3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C9 | |
220 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
117 conjugacy classes — view the full character table.
| maximal subgroup of | 3−1+2×#5933 |
| maximal normal subgroup of | 3−1+2×#5933 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30578);