Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C215 |
| direct product | C24×C3×D32 |
| unambiguous semidirect product | (C3×D3)⋊(C25×C3) |
| Fitting, socle | C24×C33 ⋊ C22 |
| derived | C32 ⋊ C26×C3 |
| Sylow | C33 ⋊ C26 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C3 | D32 |
| commutator, cosocle, hypocenter | C32 | C26×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C33 | C22 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C24×C3 | |
24978 conjugacy classes of subgroups; only the 7280 normal subgroups are shown.
This group has 24978 conjugacy classes of subgroups — too many to draw. Only its 7280 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
432 conjugacy classes. The character table is too large to display.
| maximal quotient of | C2×C3×#10596675 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47928);