Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| derived, Hall | C7×C127 ⋊ C32×C4 |
| Hall | ΓL(1,𝔽8)×C127⋊C3 ⋊ C4 |
| Hall | Dic889 ⋊ C32 |
| Sylow | C127 ⋊ C3×ΓL(1,𝔽8)⋊C4 |
| Sylow | C7 ⋊ C3×(C127⋊C3)⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽8)⋊(D127⋊C3) |
| commutator, hypocenter | C7×C127 | C32×C4 |
| Fitting, socle, cosocle | C2×C7×C127 | C2×C32 |
| prime radical | C2×ΓL(1,𝔽8)×C127⋊C3 | C2 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 ◁ C2 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 3556 |
Polycyclic presentation:
2,3^2,7,127,2|a^2=f,d^a=d^6,e^a=e^126,e^b=e^107,d^c=d^2252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 92);