Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C33 |
| direct product | #41×C3⋊(C3⋊D3) |
| Fitting | C33×#41 ⋊ C2 |
| Sylow | C33 ⋊ C2×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C33 | C24 |
| Frattini | C2 | C23×C3⋊(C3⋊D3) |
| Fitting | C33×#41 | C2 |
| hypercenter | #41 | C3⋊(C3⋊D3) |
| hypocenter | C33 | C2×#41 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ #41 | |
2296 conjugacy classes of subgroups; only the 529 normal subgroups are shown.
This group has 2296 conjugacy classes of subgroups — too many to draw. Only its 529 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
150 conjugacy classes — view the full character table.
| maximal subgroup of | #13161354, #13161364, #104×C3⋊(C3⋊D3), … |
| maximal normal subgroup of | #13161354, #13161364, #104×C3⋊(C3⋊D3), … |
| maximal quotient of | #13179642, #13179643, #13179644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4650);