Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊(#11765919⋊C7) |
| unambiguous semidirect product | (C22⋊#11765919)⋊C7 |
| Fitting, socle, derived, Hall | C22×C127 ⋊ C7×C9 |
| Hall | C22⋊#11765919 ⋊ C7 |
| Hall | C22×C127⋊C7 ⋊ C9 |
| Hall | C127⋊C7 ⋊ C22⋊C9 |
| Sylow | C127 ⋊ C7×C22⋊C9 |
| Sylow | C22 ⋊ #11765919⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C22×C127 | C7×C9 |
| prime radical | C22×(C127⋊C3)⋊C7 | C3 |
| cosocle | C22×C127⋊C3 | C3×C7 |
| LCS | G ▷ C22×C127 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| transitive degree | μt(G) = 508 |
Polycyclic presentation:
3,7,3,2^2,127|a^3=c,d^a=de,e^a=d,f^a=f^68,f^b=f^4,f^c=f^10792 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 155);