Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | SL(2,𝔽3)×(C22⋊C9)⋊C2 |
| derived | Q8×C22⋊C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8×C22⋊C9 | C2×C3 |
| Frattini | C2×C3 | A4×S4 |
| Fitting | C22×C3×Q8 | C3×D3 |
| socle | C23×C3 | D3×A4 |
| LCS | G ▷ Q8×C22⋊C9 | |
| UCS | C1 ◁ C2 | |
234 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
63 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30096);