Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×#11722371 |
| Fitting | C22×Q8×C9 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×C9⋊S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C23 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C22×Q8×C9 | D3 |
| socle | C22×C3 | C2×C3⋊S4 |
| hypercenter | C23 | C9⋊S4 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C23 | |
631 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 631 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30520);