Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (Q8⋊C9)⋊C16 |
| derived | Q8⋊C9 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | (C22⋊C9)⋊C2 |
| commutator, hypocenter | Q8⋊C9 | C16 |
| Frattini | C2×C3×C8 | S4 |
| Fitting | C3×C8×Q8 | D3 |
| socle, prime radical | C22×C3 | A4⋊C8 |
| cosocle | C8×Q8⋊C9 | C2 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2×C8 | |
89 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1152 |
Polycyclic presentation:
2^4,3^2,2^3|a^2=b,b^2=c,c^2=d,e^3=f,g^2=i,h^2=i,e^a=e^2f^2,f^a=f^2,g^a=gh,h^a=hi,g^e=h,h^e=ghi,h^g=hi120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153943);