Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #11765917⋊Dic7 |
| derived, Hall | C7×C127 ⋊ C4×C9 |
| Hall | C7×#11765917 ⋊ C4 |
| Hall | Dic889 ⋊ C9 |
| Hall | #11765917 ⋊ Dic7 |
| Sylow | C127 ⋊ C9×Dic7 |
| Sylow | C7 ⋊ #11765917⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | (C127⋊C3)⋊D7 |
| commutator, hypocenter | C7×C127 | C4×C9 |
| Fitting, socle, prime radical, cosocle | C2×C3×C7×C127 | C2×C3 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 ◁ C2×C3 | |
36 conjugacy classes of subgroups.
| transitive degree | μt(G) = 32004 |
Polycyclic presentation:
2,3,7,127,2,3|a^2=e,b^3=f,c^a=c^6,d^a=d^126,d^b=d^107972 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 26);