Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C3 |
| direct product | C3×#64367 |
| Fitting | C33×#41 ⋊ C2 |
| Sylow | C33 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C23×C3⋊D3 |
| commutator, cosocle | C2×C32 | C24×C3 |
| Frattini | C2 | C23×C3×C3⋊D3 |
| Fitting | C33×#41 | C2 |
| socle | C2×C33 | C24 |
| hypercenter | C3×#41 | C3⋊D3 |
| hypocenter | C32 | C3×2−1+4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C3 ◁ C3×#41 | |
1036 conjugacy classes of subgroups; only the 306 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 306 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
171 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#10591668, C3×#10591686, #13161390, … |
| maximal normal subgroup of | C3×#10591668, C3×#10591686, #13161390, … |
| maximal quotient of | C3×#10595987, C3×#10596002, C3×#10596004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4637);