Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of A4×#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24×C3 |
| direct product | C22×A4×#157 |
| Fitting, socle | C24×C32 ⋊ C3×C4 |
| derived | C22×C32 ⋊ C22×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4×#157 |
| commutator, hypocenter | C22×C32 | C22×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C24×C32 | C3×C4 |
| prime radical | C22×A4×C3⋊D3 | C2 |
| cosocle | C22×C3⋊D3 | C23×C3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 ◁ C22 | |
1030 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47891);