Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#63692 |
| derived | Q8⋊C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C22×C3 |
| Frattini | C2×C3 | C2×C3×S4 |
| Fitting | C32×#41 | D3 |
| socle | C2×C32 | C2×S4 |
| hypercenter | C3×C4 | (C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
157 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
81 conjugacy classes — view the full character table.
| maximal subgroup of | #13146360, #13146362, #13146367, … |
| maximal normal subgroup of | #13146360, #13146362, #13146367, … |
| maximal quotient of | C3×#10589493, C3×C4⋊#3689, C3×#10589496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2323);