Finite Group DatabasebetaFinite Group Databasebeta| order | 7644 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C7 |
| direct product | C3×C7⋊(C7⋊#261) |
| Fitting, socle, Hall | C3×C72×C13 ⋊ C4 |
| derived, Hall | C72×C13 ⋊ C3×C4 |
| Hall | C3×C72 ⋊ #261 |
| Sylow | C72 ⋊ C3×#261 |
| Hall | C3×C13 ⋊ C7⋊Dic7 |
| Sylow | C13 ⋊ C3×C7⋊Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C7⋊(C7⋊#261) |
| commutator, hypocenter | C72×C13 | C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C3×C72×C13 | C4 |
| LCS | G ▷ C72×C13 | |
| UCS | C1 ◁ C3 | |
120 conjugacy classes of subgroups.
597 conjugacy classes — view the full character table.
| maximal subgroup of | #37524412, #37524413, #37524416, … |
| maximal normal subgroup of | #37524412, #37524413, #37524416, … |
| maximal quotient of | C9×C7⋊(C7⋊#261), C32×C7⋊(C7⋊#261), C3×C3⋊(C7⋊(C7⋊#261)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7644, 316);