Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#64320 |
| Fitting | C33×Q8 ⋊ C22 |
| Sylow | C33 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×D32 |
| commutator, cosocle | C2×C32 | C24×C3 |
| Frattini | C2 | C22×C3×D32 |
| Fitting | C33×Q8 | C22 |
| socle | C2×C33 | C24 |
| hypercenter | C3×Q8 | D32 |
| hypocenter | C32 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C3 ◁ C3×Q8 | |
796 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 796 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
126 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#10593473, C3×#10593475, #13167255, … |
| maximal normal subgroup of | C3×#10593473, C3×#10593475, #13167255, … |
| maximal quotient of | C3×#10594505, C3×#10594506, C3×#10594507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4515);