Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29×C3 |
| direct product | C23×C3×PSU(3,𝔽2) |
| Fitting, socle | C23×C33 ⋊ Q8 |
| Sylow | C33 ⋊ C23×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | PSU(3,𝔽2) |
| commutator, cosocle | C3⋊D3 | C25×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C33 | Q8 |
| prime radical | C23×C3×C3⋊D3 | C22 |
| hypocenter | C32 | C23×C3×Q8 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C23×C3 | |
1866 conjugacy classes of subgroups; only the 882 normal subgroups are shown.
This group has 1866 conjugacy classes of subgroups — too many to draw. Only its 882 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47913);