Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #3774 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C22×C3×#3774 |
| Sylow | C33 ⋊ C22×D8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | GO+(4,𝔽2) |
| commutator, cosocle | C3⋊Dic3 | C24×C3 |
| Frattini | C2 | C22×C3×GO+(4,𝔽2) |
| Fitting, socle | C23×C33 | D4 |
| hypocenter | C32 | C22×C3×D8 |
| LCS | G ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C23×C3 | |
1416 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47111);