Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #63710⋊C3 |
| Fitting | C2×D4×3−1+2 ⋊ C2 |
| Sylow | 3−1+2 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×AGL(1,ℤ/9) |
| commutator, cosocle | C2×C9 | C24×C3 |
| Frattini, socle | C2×C3 | C23×C3×D3 |
| Fitting | C2×D4×3−1+2 | C2 |
| hypercenter | C2×D4 | AGL(1,ℤ/9) |
| hypocenter | C9 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
525 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 525 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3,2,3^2|g^3=h,d^a=df,g^a=g^2h^2,h^a=h^2,c^b=cf,g^e=gh^291 conjugacy classes — view the full character table.
| maximal subgroup of | #10588421⋊C3, #10588422⋊C3, #10588426⋊C3, … |
| maximal normal subgroup of | #10588421⋊C3, #10588422⋊C3, #10588426⋊C3, … |
| maximal quotient of | #10589585⋊C3, #10589588⋊C3, #10589592⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2585);