Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D9×#126⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | D9×#126⋊C3 |
| Fitting | C9×#126 ⋊ C2×C3 |
| derived | C42×C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C42×C9 | C22×C3 |
| Frattini, socle | C22×C3 | C2×D3×A4 |
| Fitting | C9×#126 | C2×C3 |
| LCS | G ▷ C42×C9 | |
| UCS | C1 | |
210 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13179);