Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊(Q8⋊C27) |
| unambiguous semidirect product | Q8⋊(C7⋊C27) |
| derived, Hall | C7×Q8 ⋊ C27 |
| Sylow | Q8 ⋊ C7⋊C27 |
| Sylow | C7 ⋊ Q8⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C9 | C22⋊ΓL(1,𝔽8) |
| commutator, hypocenter | C7×Q8 | C27 |
| Fitting, cosocle | C7×Q8×C9 | C3 |
| socle, prime radical | C2×C3×C7 | C22⋊C9 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C2×C9 | |
30 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1512 |
Polycyclic presentation:
3^3,2^2,7,2|a^3=b,b^3=c,d^2=g,e^2=g,d^a=de,e^a=d,f^a=f^2,e^d=eg153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 23);