Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #41×3−1+2 ⋊ C2 |
| Sylow | 3−1+2 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×AGL(1,ℤ/9) |
| commutator, cosocle | C2×C9 | C24×C3 |
| Frattini, socle | C2×C3 | C23×C3×D3 |
| Fitting | #41×3−1+2 | C2 |
| hypercenter | #41 | AGL(1,ℤ/9) |
| hypocenter | C9 | C3×2−1+4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ #41 | |
458 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 458 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) = 144 |
Polycyclic presentation:
2^4,3,2,3^2|b^2=f,c^2=f,d^2=f,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 | #10588506⋊C3, #10588507⋊C3, #10588881⋊C3, … |
| maximal normal subgroup of | #10588506⋊C3, #10588507⋊C3, #10588881⋊C3, … |
| maximal quotient of | #10589618⋊C3, #10589633⋊C3, #10589635⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2609);