Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C3×#10596623 |
| Fitting | C32×H(5,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×D3×A4 |
| commutator, cosocle, hypocenter | C3×Q8 | C23×C32 |
| Frattini | C2 | C22×C3×D3×A4 |
| Fitting | C32×H(5,𝔽2) | C2×C3 |
| socle | C2×C32 | C23×A4 |
| hypercenter | C3×Q8 | D3×A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C3×Q8 | |
1036 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47514);