Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
No aliases for this group.
| Sylow | C33 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×C3⋊D3 |
| commutator, socle, cosocle | C2×C33 | C24 |
| Fitting | C33×Q8 | C22 |
| hypercenter | Q8 | D3×C3⋊D3 |
| hypocenter | C33 | 2−1+4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
1036 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 200 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|c^2=e,d^2=e,b^a=be,h^a=h^2,f^b=f^2,g^b=g^2,d^c=de87 conjugacy classes — view the full character table.
| maximal subgroup of | #13171691, #13171696, #13171715, … |
| maximal normal subgroup of | #13171691, #13171696, #13171715, … |
| maximal quotient of | #13175972, #13175973, #13175975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4558);