Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (#11765919⋊C4)⋊C7 |
| unambiguous semidirect product | (#11765919⋊C7)⋊C4 |
| derived, Sylow | C127 ⋊ C4×C7×C9 |
| Hall | #11765919⋊C7 ⋊ C4 |
| Hall | #11765919⋊C4 ⋊ C7 |
| Hall | Dic127⋊C7 ⋊ C9 |
| Hall | #11765919 ⋊ C4×C7 |
| Hall | C127⋊C7 ⋊ C4×C9 |
| Hall | Dic127 ⋊ C7×C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AGL(1,𝔽127) |
| commutator, hypocenter | C127 | C4×C7×C9 |
| Fitting, socle | C2×C127 | C2×C7×C9 |
| prime radical | C2×(C127⋊C3)⋊C7 | C2×C3 |
| cosocle | C2×C127⋊C3 | C2×C3×C7 |
| LCS | G ▷ C127 | |
| UCS | C1 ◁ C2 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 508 |
Polycyclic presentation:
2,3,7,3,127,2|a^2=f,b^3=d,e^a=e^126,e^b=e^68,e^c=e^4,e^d=e^107254 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 81);