Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C2×C3 |
| direct product | C22×C3×C7⋊C9 |
| derived, Sylow | C7 ⋊ C22×C3×C9 |
| Hall | C22×C7 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C22×C3×C9 |
| Frattini | C3 | C22×C3×ΓL(1,𝔽8) |
| Fitting, socle, prime radical | C22×C32×C7 | C3 |
| cosocle | C3×C7 | C22×C32 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C32 | |
100 conjugacy classes of subgroups.
| transitive degree | μt(G) = 756 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 104);