Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3×C9 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D3×D9 |
| commutator, cosocle | C2×C3×C9 | C24 |
| Frattini | C2×C3 | C22×D32 |
| Fitting | C3×Q8×C9 | C22 |
| socle | C2×C32 | C23×D3 |
| hypercenter | Q8 | D3×D9 |
| hypocenter | C3×C9 | 2−1+4 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
476 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 131 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=e,c^2=e,d^2=e,f^3=h,d^a=de,g^a=g^2,c^b=ce,f^b=f^2h^2,h^b=h^2,d^c=de87 conjugacy classes — view the full character table.
| maximal subgroup of | #13149945, #13149951, #13149954, … |
| maximal normal subgroup of | #13149945, #13149951, #13149954, … |
| maximal quotient of | #13148742, #13148745, #13148748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2475);