Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#5936 |
| derived | C22⋊A4 ⋊ C4×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | #1020 |
| commutator, hypocenter | C22⋊A4 | C4×C9 |
| Frattini | C2×C3 | C3×#1020 |
| Fitting | C25×C9 | D3 |
| socle | C25×C3 | C3×D3 |
| prime radical, cosocle | C2×C3×C22⋊A4 | C2×C3 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2×C9 | |
478 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30581);