Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of D3×AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| Fitting, socle | C32×C7 ⋊ C22×C3 |
| derived | C3×C7 ⋊ C22×C32 |
| Hall | C32×ΓL(1,𝔽8) ⋊ C22 |
| Sylow | C7 ⋊ C2×C32×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D3×AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C3×C7 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C32×C7 | C22×C3 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C3 | |
152 conjugacy classes of subgroups.
| transitive degree | μt(G) = 42 |
63 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 152);