Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #11765919⋊Dic7 |
| derived, Hall | C7×C127 ⋊ C4×C9 |
| Hall | C7×#11765919 ⋊ C4 |
| Hall | Dic889 ⋊ C9 |
| Hall | #11765919 ⋊ Dic7 |
| Sylow | C127 ⋊ C9×Dic7 |
| Sylow | C7 ⋊ #11765919⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #11765919⋊D7 |
| commutator, hypocenter | C7×C127 | C4×C9 |
| Fitting, socle | C2×C7×C127 | C2×C9 |
| prime radical, cosocle | C2×C7×C127⋊C3 | C2×C3 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 ◁ C2 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 3556 |
Polycyclic presentation:
2,3^2,7,127,2|a^2=f,b^3=c,d^a=d^6,e^a=e^126,e^b=e^68,e^c=e^107188 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 97);