Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#63958 |
| Sylow | C33 ⋊ #121 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C2×D32 |
| commutator, cosocle | C22×C32 | C23×C3 |
| Frattini | C22 | C2×C3×D32 |
| Fitting | C23×C33 | C22 |
| socle | C22×C33 | C23 |
| hypercenter | C23×C3 | D32 |
| hypocenter | C32 | C3×#121 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 | |
550 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 550 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
126 conjugacy classes — view the full character table.
| maximal subgroup of | #13171752, #13171753, #13171763, … |
| maximal normal subgroup of | #13171752, #13171753, #13171763, … |
| maximal quotient of | C3×#10591204, C3×#10591312, C3×#10591281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3250);