Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#11722394 |
| Fitting | C2×C32×#41 ⋊ D3 |
| derived | Q8⋊H(3,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C3⋊S4 |
| commutator, cosocle, hypocenter | Q8⋊H(3,𝔽3) | C23 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C2×C32×#41 | D3 |
| hypercenter | C2×C3×C4 | C3⋊S4 |
| LCS | G ▷ Q8⋊H(3,𝔽3) | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C4 | |
828 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 828 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
98 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13147078 |
| maximal normal subgroup of | C2×#13147078 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30561);