Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of A4×GO+(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | A4×GO+(4,𝔽2) |
| unambiguous semidirect product | #157⋊(C2×A4) |
| unambiguous semidirect product | D32⋊(C2×A4) |
| unambiguous semidirect product | (C22×GO+(4,𝔽2))⋊C3 |
| Fitting, socle | C22×C32 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C22×C3⋊D3 | C22×C3 |
| Fitting, socle, hypocenter | C22×C32 | C3×D4 |
| LCS | G ▷ C22×C3⋊D3 ▷ C22×C32 | |
| UCS | C1 | |
226 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
36 conjugacy classes — view the full character table.
| maximal subgroup of | #13181779, A4×AΓL(1,𝔽9), S4×GO+(4,𝔽2), … |
| maximal normal subgroup of | #13181779, A4×AΓL(1,𝔽9), S4×GO+(4,𝔽2), … |
| maximal quotient of | D32⋊(#41⋊C3), #157⋊(#41⋊C3), #10596520⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4671);