Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C3⋊AGL(1,𝔽7))⋊D127 |
| unambiguous semidirect product | (ΓL(1,𝔽8)⋊D127)⋊D3 |
| Fitting, socle, derived | C3×C7×C127 ⋊ C22×C3 |
| Hall | C3×ΓL(1,𝔽8)×C127 ⋊ C22 |
| Hall | C7×C127 ⋊ C2×C3×D3 |
| Sylow | C127 ⋊ D3×AGL(1,𝔽7) |
| Hall | C3×ΓL(1,𝔽8) ⋊ C2×D127 |
| Sylow | C7 ⋊ C3×D3×D127 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C127 | C22×C3 |
| LCS | G ▷ C3×C7×C127 | |
| UCS | C1 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2667 |
Polycyclic presentation:
2^2,3^2,7,127|e^a=e^6,f^a=f^126,d^b=d^2,e^b=e^6,e^c=e^2777 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 211);