Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×#11722264 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×(C22⋊C9)⋊C2 |
| commutator, cosocle | C2×C3×C22⋊C9 | C23 |
| Frattini | C2×C3 | C2×D3×S4 |
| Fitting, socle | C24×C32 | C2×D3 |
| hypocenter | C3×C22⋊C9 | C2×D4 |
| LCS | G ▷ C2×C3×C22⋊C9 ▷ C3×C22⋊C9 | |
| UCS | C1 ◁ C22 | |
955 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 955 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30305);