Finite Group DatabasebetaFinite Group Databasebeta| order | 7644 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×D7×D39 |
| unambiguous semidirect product | (C3×D7)⋊(C7×D13) |
| unambiguous semidirect product | (C13×D7)⋊(D3×C7) |
| Fitting, socle, Hall | C3×C72×C13 ⋊ C22 |
| derived | C3×C7×C13 ⋊ C22×C7 |
| Hall | C72×C13 ⋊ C2×D3 |
| Hall | C3×C72 ⋊ C2×D13 |
| Sylow | C72 ⋊ C2×D39 |
| Hall | C3×C13 ⋊ C2×C7×D7 |
| Sylow | C13 ⋊ D3×C7×D7 |
| Sylow | C3 ⋊ C7×D7×D13 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | D7×D39 |
| commutator, hypocenter | C3×C7×C13 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C3×C72×C13 | C22 |
| LCS | G ▷ C3×C7×C13 | |
| UCS | C1 ◁ C7 | |
104 conjugacy classes of subgroups.
735 conjugacy classes — view the full character table.
| maximal subgroup of | C7×D7×D117, D39×#64418, C7×#10587721⋊D3, … |
| maximal normal subgroup of | C7×#10587721⋊D3, C7×#10587720⋊D3, C7×AGL(1,𝔽7)×D39, … |
| maximal quotient of | C7×D7×D117, C3×C7×D7×D39, C7×D21×D39, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7644, 329);