Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of (C3⋊D3)⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×(C3⋊D3)⋊ΓL(1,𝔽8) |
| unambiguous semidirect product | (C2×C3⋊D3)⋊ΓL(1,𝔽8) |
| Fitting | C2×C32×C7 ⋊ C2×C3 |
| derived | C32×C7 ⋊ C22×C3 |
| Hall | C7⋊H(3,𝔽3) ⋊ C22 |
| Sylow | C7 ⋊ C2×(C3⋊D3)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C3⋊D3)⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C32×C7 | C22×C3 |
| Frattini | C3 | C2×D3×ΓL(1,𝔽8) |
| Fitting | C2×C32×C7 | C2×C3 |
| socle | C2×C3×C7 | C3×D3 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 80);