Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D32×Dic6 |
| Sylow | C33 ⋊ C22×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D33 |
| commutator, socle, cosocle | C2×C33 | C24 |
| Fitting | C33×C4 | C23 |
| prime radical | C2×C3×D32 | C22 |
| hypercenter | C4 | D33 |
| hypocenter | C33 | C22×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
750 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 750 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
81 conjugacy classes — view the full character table.
| maximal subgroup of | #13165196, #13165198, #13165212, … |
| maximal normal subgroup of | #13165196, #13165198, #13165212, … |
| maximal quotient of | D3×Dic3×Dic6, D3×#63845, #13167878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4337);