Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C32 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C3⋊(C3⋊C32))⋊C3 |
| derived | C32 ⋊ C3×C32 |
| Sylow | H(3,𝔽3) ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C3×C32 |
| Frattini | C3×C16 | C3×D3 |
| Fitting | C16×H(3,𝔽3) | C2 |
| socle | C2×C3 | C3×C3⋊C16 |
| prime radical | C2×H(3,𝔽3) | C16 |
| cosocle | C32×C16 | C2×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C16 | |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2,3,2^4,3^2|a^2=c,c^2=d,d^2=e,e^2=f,g^a=g^2,h^a=h^2,g^b=gh160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 66);