Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D3×#64188 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24 |
| direct product | D3×#64188 |
| derived | C3×C22⋊(C22⋊C9) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C22⋊(C22⋊C9) | C22 |
| Frattini | C3 | D3×#1020 |
| Fitting, socle | C24×C32 | C2×D3 |
| LCS | G ▷ C3×C22⋊(C22⋊C9) | |
| UCS | C1 | |
954 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 954 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46118);