Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D33 |
| commutator, socle, cosocle | C2×C33 | C24 |
| Fitting | C22×C33 | C23 |
| hypercenter | C22 | D33 |
| hypocenter | C33 | 2−1+4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C22 | |
718 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 718 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^5,3^3|b^2=e,c^2=e,d^a=de,h^a=h^2,c^b=ce,g^b=g^2,d^c=de,f^c=f^269 conjugacy classes — view the full character table.
| maximal subgroup of | #13165561, #13165562, #13165565, … |
| maximal normal subgroup of | #13165561, #13165562, #13165565, … |
| maximal quotient of | #13168162, #13168163, #13168189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4394);