Finite Group DatabasebetaFinite Group Databasebeta| order | 7644 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | Dic7×#63270 |
| derived, Hall | C72×C13 ⋊ C3×C4 |
| Hall | C7×C13×Dic7 ⋊ C3 |
| Hall | C7×#63270 ⋊ C4 |
| Hall | C7×Dic7 ⋊ C13⋊C3 |
| Sylow | C72 ⋊ C4×C13⋊C3 |
| Sylow | C13 ⋊ ΓL(1,𝔽8)×Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×#63270 |
| commutator, hypocenter | C72×C13 | C3×C4 |
| Fitting, socle | C2×C72×C13 | C2×C3 |
| LCS | G ▷ C72×C13 | |
| UCS | C1 ◁ C2 | |
52 conjugacy classes of subgroups.
330 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)⋊C4×#63270, #37524657, #37524660, … |
| maximal normal subgroup of | ΓL(1,𝔽8)⋊C4×#63270, #37524657, #37524660, … |
| maximal quotient of | Dic7×#11717941, C3×Dic7×#63270, Dic21×#63270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7644, 126);