Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #6290 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C3 |
| direct product | C23×#6293 |
| Sylow | C33 ⋊ C23×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | D3×C3⋊D3 |
| commutator, cosocle | C2×C33 | C25 |
| Frattini | C2 | C23×D3×C3⋊D3 |
| Fitting, socle, prime radical | C24×C33 | C22 |
| hypocenter | C33 | C23×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C24 | |
6044 conjugacy classes of subgroups; only the 1468 normal subgroups are shown.
This group has 6044 conjugacy classes of subgroups — too many to draw. Only its 1468 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
264 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47722);