Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C9×#994 |
| Fitting | C9×H(5,𝔽2) ⋊ C3 |
| Sylow | H(5,𝔽2) ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | C22×A4 |
| commutator, hypocenter | Q8 | C22×C3×C9 |
| Frattini, socle | C2×C3 | C22×C3×A4 |
| Fitting | C9×H(5,𝔽2) | C3 |
| prime radical | C3×#994 | C3 |
| cosocle | C3×Q8 | C22×C32 |
| hypercenter | Q8×C9 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C9 ◁ Q8×C9 | |
182 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
171 conjugacy classes — view the full character table.
| maximal subgroup of | C9×#5428, C9×#5429, #13146984, … |
| maximal normal subgroup of | C9×#5428, C9×#5429, #13146984, … |
| maximal quotient of | C9×#520⋊C3, C9×C4⋊(#41⋊C3), Q8×C9×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2523);