Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#10593099 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C23×C3 |
| Frattini | C2×C3 | C22×C3×S4 |
| Fitting | C32×H(5,𝔽2) | D3 |
| socle | C2×C32 | C22×S4 |
| hypercenter | C3×Q8 | (C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2×C3 ◁ C3×Q8 | |
536 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
126 conjugacy classes — view the full character table.
| maximal subgroup of | #72252981, C3×#13164432, #72252993 |
| maximal normal subgroup of | #72252981, C3×#13164432, #72252993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30494);