Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
| direct product | C2×#89018 |
| Sylow | C33 ⋊ C2×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×C3⋊D3 |
| commutator, cosocle | C2×C33 | C24 |
| Frattini | C2 | C22×D3×C3⋊D3 |
| Fitting | C2×C33×C4 | C22 |
| socle | C22×C33 | C23 |
| hypercenter | C2×C4 | D3×C3⋊D3 |
| hypocenter | C33 | C2×#41 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
1200 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 224 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | #13155343, #13155376, #13155378, … |
| maximal normal subgroup of | #13155343, #13155376, #13155378, … |
| maximal quotient of | #13175139, #13175142, #13175144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4534);