Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊#22547883 |
| Fitting, socle, derived, Hall | C22×C7×C127 ⋊ C9 |
| Hall | C7×C127 ⋊ C22⋊C9 |
| Hall | C22×C127 ⋊ C7⋊C9 |
| Sylow | C127 ⋊ C22⋊(C7⋊C9) |
| Hall | C22×C7 ⋊ #11765919 |
| Sylow | C7 ⋊ C22⋊#11765919 |
| Sylow | C22 ⋊ #22547883 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C22×C7×C127 | C9 |
| prime radical, cosocle | C22×C7×C127⋊C3 | C3 |
| LCS | G ▷ C22×C7×C127 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| transitive degree | μt(G) = 3556 |
Polycyclic presentation:
3^2,2^2,7,127|a^3=b,c^a=d,d^a=cd,e^a=e^2,f^a=f^68,f^b=f^107428 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 222);