Finite Group DatabasebetaFinite Group Databasebeta| order | 7644 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×D21⋊D7 |
| unambiguous semidirect product | D21⋊(C13×D7) |
| Fitting, socle, Hall | C3×C72×C13 ⋊ C22 |
| derived, Hall | C3×C72 ⋊ C22×C13 |
| Hall | C72×C13 ⋊ C2×D3 |
| Sylow | C72 ⋊ C2×D3×C13 |
| Hall | C3×C13 ⋊ D72 |
| Sylow | C3 ⋊ C13×D72 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C13 | D21⋊D7 |
| commutator, hypocenter | C3×C72 | C22×C13 |
| Frattini | C1 | G |
| Fitting, socle | C3×C72×C13 | C22 |
| LCS | G ▷ C3×C72 | |
| UCS | C1 ◁ C13 | |
104 conjugacy classes of subgroups.
663 conjugacy classes — view the full character table.
| maximal subgroup of | C13×#11725071⋊C2, C13×D63⋊D7, C13×#13183625, … |
| maximal normal subgroup of | C13×(C3⋊AGL(1,𝔽7))⋊D7, #37525228, #37525229, … |
| maximal quotient of | C13×D63⋊D7, C3×C13×D21⋊D7, C13×D21⋊D21, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7644, 342);