Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C33 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊AGL(1,𝔽9) |
| commutator, socle | C2×C33 | C2×C8 |
| Fitting, prime radical | C22×C33 | C8 |
| cosocle | C3×#622 | C22 |
| hypercenter | C22 | C3⋊AGL(1,𝔽9) |
| hypocenter | C33 | M32 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C22 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^5,3^3|a^2=c,c^2=d,d^2=e,b^a=be,f^a=f^2g^2,g^a=fg^2,h^a=h^2,f^c=g^2,g^c=f,f^d=f^2,g^d=g^248 conjugacy classes — view the full character table.
| maximal subgroup of | #13165066, #13165067, #13165103, … |
| maximal normal subgroup of | #13165066, #13165067, #13165103, … |
| maximal quotient of | #13166216, #13166219, #13166227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4209);