Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of C3⋊((C22⋊C9)⋊C2) |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊((C22⋊C9)⋊C8) |
| derived | C3×C22⋊C9 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊((C22⋊C9)⋊C2) |
| commutator, hypocenter | C3×C22⋊C9 | C8 |
| Frattini | C3×C4 | C3⋊S4 |
| Fitting | C22×C32×C4 | D3 |
| socle, prime radical | C23×C32 | Dic3 |
| cosocle | C3×C4×C22⋊C9 | C2 |
| LCS | G ▷ C3×C22⋊C9 | |
| UCS | C1 ◁ C4 | |
140 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^3,3^3,2^2|a^2=b,b^2=c,d^3=f^2,d^a=d^2f,e^a=e^2,f^a=f^2,g^a=h,h^a=g,g^d=h,h^d=gh84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 702);