Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C24×C3 |
| direct product | C23×C9×A4 |
| Fitting | C25×C9 ⋊ C3 |
| derived | C22 ⋊ C23×C3×C9 |
| Sylow | C25 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C9 | A4 |
| commutator, hypocenter | C22 | C23×C3×C9 |
| Frattini | C3 | C23×C3×A4 |
| Fitting | C25×C9 | C3 |
| socle | C25×C3 | C32 |
| prime radical | C23×C3×A4 | C3 |
| cosocle | C22×C3 | C23×C32 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C23×C9 | |
662 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C9×#5432, #13147034, A4×#3676, … |
| maximal normal subgroup of | C9×#5432, #13147034, A4×#3676, … |
| maximal quotient of | C22×C4×C9×A4, C2×D4×C9×A4, C23×C9×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4034);