Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C33 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D3×C3⋊D3 |
| commutator, socle | C2×C33 | C2×C8 |
| Fitting, cosocle | C33×C8 | C22 |
| prime radical | C2×C3×C3⋊D3 | C8 |
| hypocenter | C33 | M32 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C8 | |
196 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^5,3^3|a^2=c,c^2=d,d^2=e,b^a=be,h^a=h^2,f^b=f^2,g^b=g^2132 conjugacy classes — view the full character table.
| maximal subgroup of | #13154733, #13154734, #13154751, … |
| maximal normal subgroup of | #13154733, #13154734, #13154751, … |
| maximal quotient of | #13145546, #13145552, #13145601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1969);