Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · 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×C3⋊D3 | C25 |
| Frattini | C1 | G |
| Fitting, socle | C23×C33 | Q8 |
| prime radical | C23×C3×C3⋊D3 | C22 |
| hypocenter | C33 | C23×Q8 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C23 | |
1866 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1866 conjugacy classes of subgroups — too many to draw. Only its 575 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47914);