Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of A4×AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C2 |
| direct product | A4×AGL(1,𝔽9) |
| unambiguous semidirect product | (C22×AGL(1,𝔽9))⋊C3 |
| Fitting, socle, derived | C22×C32 ⋊ C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C22×C32 | C3×C8 |
| prime radical | A4×C3⋊D3 | C4 |
| cosocle | C22×#157 | C2×C3 |
| LCS | G ▷ C22×C32 | |
| UCS | C1 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
36 conjugacy classes — view the full character table.
| maximal subgroup of | S4×AGL(1,𝔽9), AGL(1,𝔽9)⋊S4, A4×AΓL(1,𝔽9), … |
| maximal normal subgroup of | S4×AGL(1,𝔽9), AGL(1,𝔽9)⋊S4, A4×AΓL(1,𝔽9), … |
| maximal quotient of | #10596487⋊C3, SL(2,𝔽3)×AGL(1,𝔽9), A4×#3771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4664);