Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊((C127⋊C3)⋊Dic7) |
| unambiguous semidirect product | (C127⋊C3)⋊Dic21 |
| derived | C3×C7×C127 ⋊ C3×C4 |
| Hall | C3×C7×C127⋊C3 ⋊ C4 |
| Hall | C3×C127⋊C3 ⋊ Dic7 |
| Hall | C7×C127 ⋊ C3×Dic3 |
| Sylow | C127 ⋊ C3×Dic21 |
| Sylow | C7 ⋊ C3⋊((C127⋊C3)⋊C4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊((C127⋊C3)⋊D7) |
| commutator, hypocenter | C3×C7×C127 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C7×C127 | C2×C3 |
| prime radical | C2×C3×C7×C127⋊C3 | C2 |
| LCS | G ▷ C3×C7×C127 | |
| UCS | C1 ◁ C2 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 10668 |
Polycyclic presentation:
2,3^2,7,127,2|a^2=f,c^a=c^2,d^a=d^6,e^a=e^126,e^b=e^107954 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 135);