Finite Group DatabasebetaFinite Group Databasebeta| order | 32004 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| Fitting, socle, derived, Hall | C7×C127 ⋊ C22×C32 |
| Hall | ΓL(1,𝔽8)×C127⋊C3 ⋊ C22 |
| Hall | D7×D127 ⋊ C32 |
| Sylow | C127 ⋊ C2×C3×AGL(1,𝔽7) |
| Sylow | C7 ⋊ C2×C3×D127⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C127 | C22×C32 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 | |
120 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 889 |
189 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32004, 163);