Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | #14581888⋊A4 |
| Fitting, socle, derived, Hall | C22×C7×C127 ⋊ C32 |
| Hall | C7×C127 ⋊ C3×A4 |
| Hall | C22×C127 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C127 ⋊ A4×ΓL(1,𝔽8) |
| Hall | C22×C7 ⋊ C3×C127⋊C3 |
| Sylow | C7 ⋊ A4×C127⋊C3 |
| Sylow | C22 ⋊ ΓL(1,𝔽8)×C127⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C7×C127 | C32 |
| LCS | G ▷ C22×C7×C127 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 3556 |
Polycyclic presentation:
3^2,2^2,7,127|e^a=e^2,f^a=f^107,c^b=cd,d^b=c,e^b=e^2524 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 229);