Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D3×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C3⋊D9)⋊C16 |
| derived, Sylow | C3×C9 ⋊ C2×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D3×D9 |
| commutator, hypocenter | C3×C9 | C2×C16 |
| Frattini | C3×C8 | D32 |
| Fitting, cosocle | C3×C8×C9 | C22 |
| socle | C2×C32 | D3×C8 |
| prime radical | C2×C3⋊D9 | C8 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C8 | |
106 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^5,3^3|a^2=c,b^2=c,c^2=d,d^2=e,f^3=h,g^a=g^2,f^b=f^2h^2,h^b=h^2144 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C16×D9, #13136961, #13136963, … |
| maximal normal subgroup of | D3×C16×D9, #13136961, #13136963, … |
| maximal quotient of | (C3⋊D9)⋊C32, #13134894, #13135012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 196);