Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C3 |
| direct product | SL(2,𝔽3)×PSU(3,𝔽2) |
| Fitting | C32×Q8 ⋊ C3×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×PSU(3,𝔽2) |
| commutator, cosocle | Q8×C3⋊D3 | C22×C3 |
| Fitting, hypocenter | C32×Q8 | C3×Q8 |
| socle | C2×C32 | Q8×A4 |
| prime radical | C3⋊D3×SL(2,𝔽3) | C22 |
| LCS | G ▷ Q8×C3⋊D3 ▷ C32×Q8 | |
| UCS | C1 ◁ C2 | |
178 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46295);