Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C33 |
No aliases for this group.
| Fitting | C2×C33×Q8 ⋊ C2 |
| Sylow | C33 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C33 | C24 |
| Fitting | C2×C33×Q8 | C2 |
| hypercenter | C2×Q8 | C3⋊(C3⋊D3) |
| hypocenter | C33 | 2−1+4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
2044 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^5,3^3|b^2=e,c^2=e,d^a=de,f^a=f^2,g^a=g^2,h^a=h^2,c^b=ce147 conjugacy classes — view the full character table.
| maximal subgroup of | #13161323, #13161324, #13161332, … |
| maximal normal subgroup of | #13161323, #13161324, #13161332, … |
| maximal quotient of | #13179626, #13179630, #13179633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4649);