Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #30630664⋊C2 |
| unambiguous semidirect product | #30630668⋊C2 |
| unambiguous semidirect product | #30630672⋊C2 |
| derived, Hall | C7×C127 ⋊ C22×C9 |
| Hall | #22547877 ⋊ C22 |
| Hall | D7×D127 ⋊ C9 |
| Sylow | C127 ⋊ C2×D7⋊C9 |
| Sylow | C7 ⋊ C2×#11765917⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | #19128625⋊C2 |
| commutator, hypocenter | C7×C127 | C22×C9 |
| Fitting, socle, cosocle | C3×C7×C127 | C22×C3 |
| prime radical | C3×D7×D127 | C3 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 ◁ C3 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 8001 |
Polycyclic presentation:
2^2,3,7,127,3|c^3=f,e^a=e^126,d^b=d^6,d^c=d^2,e^c=e^19357 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 41);