Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C128 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C9⋊C128 |
| derived, Sylow | C9 ⋊ C128 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C64 | D9 |
| commutator, hypocenter | C9 | C128 |
| Frattini | C3×C64 | D3 |
| Fitting, cosocle | C9×C64 | C2 |
| socle, prime radical | C2×C3 | C3⋊C64 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C64 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 1152 |
Polycyclic presentation:
2^7,3^2|a^2=b,b^2=c,c^2=d,d^2=e,e^2=f,f^2=g,h^3=i,h^a=h^2i^2,i^a=i^2384 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4657);