Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | SL(2,𝔽3)×Dic9 |
| derived | Q8×C9 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4×D9 |
| commutator, hypocenter | Q8×C9 | C3×C4 |
| Frattini, socle | C22×C3 | D3×A4 |
| Fitting, cosocle | C2×Q8×C9 | C2×C3 |
| prime radical | C2×C3×SL(2,𝔽3) | D3 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C22 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | Dic9×#226, #13146207, #13146212, … |
| maximal normal subgroup of | Dic9×#226, #13146207, #13146212, … |
| maximal quotient of | SL(2,𝔽3)×C9⋊C8, C2×SL(2,𝔽3)×Dic9 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 718);