Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #30630682⋊C2 |
| unambiguous semidirect product | #30630684⋊C2 |
| unambiguous semidirect product | #30630688⋊C2 |
| Fitting, socle, derived, Hall | C7×C127 ⋊ C22×C9 |
| Hall | #22547883 ⋊ C22 |
| Hall | D7×D127 ⋊ C9 |
| Sylow | C127 ⋊ C2×D7⋊C9 |
| Sylow | C7 ⋊ C2×#11765919⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C127 | C22×C9 |
| prime radical | D7×D127⋊C3 | C3 |
| cosocle | C7×C127⋊C3 | C22×C3 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 889 |
Polycyclic presentation:
2^2,3^2,7,127|c^3=d,f^a=f^126,e^b=e^6,e^c=e^2,f^c=f^68,f^d=f^10777 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 161);