Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C127⋊C7)⋊#157 |
| unambiguous semidirect product | (C127⋊#157)⋊C7 |
| Fitting, socle, derived, Hall | C32×C127 ⋊ C4×C7 |
| Hall | C32×C127⋊C7 ⋊ C4 |
| Hall | C127⋊#157 ⋊ C7 |
| Hall | C127⋊C7 ⋊ #157 |
| Sylow | C127 ⋊ C7×#157 |
| Sylow | C32 ⋊ Dic127⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C127 | C4×C7 |
| prime radical | C3⋊D3×C127⋊C7 | C2 |
| cosocle | C3⋊D3×C127 | C2×C7 |
| LCS | G ▷ C32×C127 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 762 |
Polycyclic presentation:
2,7,2,3^2,127|a^2=c,d^a=de,e^a=de^2,f^a=f^126,f^b=f^4,d^c=d^2,e^c=e^296 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 165);