Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of A42 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×A4×C22⋊C9 |
| unambiguous semidirect product | (C23×A4)⋊C9 |
| derived | C24 ⋊ C2×C3×C9 |
| Sylow | C25 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | A42 |
| commutator, hypocenter | C24 | C2×C3×C9 |
| Frattini | C3 | C2×A42 |
| Fitting, socle | C25×C3 | C32 |
| prime radical | C23×C3×A4 | C3 |
| cosocle | C24×C3 | C2×C32 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2×C3 | |
232 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4001);