Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
| direct product | GO+(4,𝔽2)×PSU(3,𝔽2) |
| Fitting, socle, Sylow | C34 ⋊ D4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C3⋊D32 | C24 |
| Fitting, socle, hypocenter | C34 | D4×Q8 |
| prime radical | C3⋊D3×GO+(4,𝔽2) | C22 |
| LCS | G ▷ C3⋊D32 ▷ C34 | |
| UCS | C1 | |
1007 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 1007 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | #72252880, GO+(4,𝔽2)×AΓL(1,𝔽9), PSU(3,𝔽2)×AΓL(1,𝔽9) |
| maximal normal subgroup of | #72252880, GO+(4,𝔽2)×AΓL(1,𝔽9), PSU(3,𝔽2)×AΓL(1,𝔽9) |
Past the SmallGroups library coverage — no SmallGroups id.