Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (#11765917⋊C4)⋊C7 |
| unambiguous semidirect product | (#11765917⋊C7)⋊C4 |
| derived, Sylow | C127 ⋊ C4×C7×C9 |
| Hall | #11765917⋊C7 ⋊ C4 |
| Hall | #11765917⋊C4 ⋊ C7 |
| Hall | Dic127⋊C7 ⋊ C9 |
| Hall | #11765917 ⋊ C4×C7 |
| Hall | C127⋊C7 ⋊ C4×C9 |
| Hall | Dic127 ⋊ C7×C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | (D127⋊C3)⋊C7 |
| commutator, hypocenter | C127 | C4×C7×C9 |
| Fitting, socle, cosocle | C2×C3×C127 | C2×C3×C7 |
| prime radical | C2×C3×C127⋊C7 | C2×C3 |
| LCS | G ▷ C127 | |
| UCS | C1 ◁ C2×C3 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 4572 |
Polycyclic presentation:
2,3,7,127,2,3|a^2=e,b^3=f,d^a=d^126,d^b=d^107,d^c=d^4270 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 30);