Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3×C9 ⋊ D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #88422 |
| commutator, Fitting, cosocle | C3×C8×C9 | C22 |
| Frattini | C3×C8 | D32 |
| socle | C2×C32 | #213 |
| hypercenter | C8 | D3×D9 |
| hypocenter | C3×C9 | D16 |
| LCS | G ▷ C3×C8×C9 ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
106 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 31 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:
2^5,3^3|b^2=cd,c^2=de,d^2=e,f^3=h,b^a=bc,c^a=cd,d^a=de,g^a=g^2,f^b=f^2h^2,h^b=h^287 conjugacy classes — view the full character table.
| maximal subgroup of | D9×D48, #13136978, #13136979, … |
| maximal normal subgroup of | D9×D48, #13136978, #13136979, … |
| maximal quotient of | #13134902, #13134904, #13134906, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 206);