Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C3⋊D3×#994 |
| Fitting | C32×H(5,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×A4×C3⋊D3 |
| commutator, cosocle, hypocenter | C32×Q8 | C23×C3 |
| Fitting | C32×H(5,𝔽2) | C2×C3 |
| socle | C2×C32 | C23×A4 |
| hypercenter | Q8 | A4×C3⋊D3 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
1336 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47534);