Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C9×Q8⋊SL(2,𝔽3) |
| Fitting | Q82×C9 ⋊ C3 |
| derived, Sylow | Q82 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C9 | C22⋊A4 |
| commutator, hypocenter | Q82 | C3×C9 |
| Frattini, socle | C22×C3 | C3×C22⋊A4 |
| Fitting | Q82×C9 | C3 |
| prime radical | C3×Q8⋊SL(2,𝔽3) | C3 |
| cosocle | C3×Q82 | C32 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C22×C9 | |
250 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18308);