Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | A4×Q8⋊C9 |
| unambiguous semidirect product | (Q8×A4)⋊C9 |
| derived, Sylow | C22×Q8 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | A42 |
| commutator, hypocenter | C22×Q8 | C3×C9 |
| Fitting, cosocle | C22×C3×Q8 | C32 |
| socle | C23×C3 | C3×A4 |
| prime radical | C2×C3×A4 | A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2×C3 | |
103 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
84 conjugacy classes — view the full character table.
| derived subgroup of | A4⋊#3688, A4⋊#3689 |
| maximal subgroup of | A4×#3688, A4⋊(#41⋊C9), A4×#3689, … |
| maximal normal subgroup of | A4×#3688, A4⋊(#41⋊C9), A4×#3689, … |
| maximal quotient of | SL(2,𝔽3)×Q8⋊C9, C2×A4×Q8⋊C9 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2243);