Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | AGL(1,𝔽7)×D381 |
| unambiguous semidirect product | D7⋊(C3×D381) |
| unambiguous semidirect product | (C3×AGL(1,𝔽7))⋊D127 |
| unambiguous semidirect product | (AGL(1,𝔽7)×C127)⋊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 ⋊ C2×C3×D381 |
| 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 |
1344 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 209);