Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of C3⋊D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C9⋊C32) |
| unambiguous semidirect product | C9⋊(C3⋊C32) |
| derived, Sylow | C3×C9 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | C3⋊D9 |
| commutator, hypocenter | C3×C9 | C32 |
| Frattini | C3×C16 | C3⋊D3 |
| Fitting, cosocle | C3×C9×C16 | C2 |
| socle, prime radical | C2×C32 | C3⋊C16 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C16 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 864 |
Polycyclic presentation:
2^5,3^3|a^2=b,b^2=c,c^2=d,d^2=e,f^3=h,f^a=f^2h^2,g^a=g^2,h^a=h^2240 conjugacy classes — view the full character table.
| maximal subgroup of | D9×C3⋊C32, #13134893, D3×C9⋊C32, … |
| maximal normal subgroup of | D9×C3⋊C32, #13134893, D3×C9⋊C32, … |
| maximal quotient of | C3⋊(C9⋊C64), C2×C3⋊(C9⋊C32) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 68);