Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C3 |
| direct product | C22×C3×PSU(3,𝔽2) |
| Fitting, socle | C22×C33 ⋊ Q8 |
| Sylow | C33 ⋊ C22×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | PSU(3,𝔽2) |
| commutator, cosocle | C3⋊D3 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C33 | Q8 |
| prime radical | C22×C3×C3⋊D3 | C22 |
| hypocenter | C32 | C22×C3×Q8 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C22×C3 | |
354 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | #13165811, #13165814, #13165816, … |
| maximal normal subgroup of | #13165811, #13165814, #13165816, … |
| maximal quotient of | C2×C3×#64243, C2×C3×C4×PSU(3,𝔽2), C2×C3×C4⋊PSU(3,𝔽2), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4702);