Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of S4×#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×S4×#157 |
| Fitting, socle | C23×C32 ⋊ C4×D3 |
| derived | C32×A4 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | S4×#157 |
| commutator, hypocenter | C32×A4 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32 | C4×D3 |
| prime radical | C2×C3⋊D3×S4 | C2 |
| cosocle | A4×C3⋊D3 | C23 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
768 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 768 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | C2×S4×PSU(3,𝔽2), C2×S4×GO+(4,𝔽2), C2×S4×AGL(1,𝔽9) |
| maximal normal subgroup of | C2×S4×PSU(3,𝔽2), C2×S4×GO+(4,𝔽2), C2×S4×AGL(1,𝔽9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47869);