Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of D3×AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| derived | C3×C7 ⋊ C22×C9 |
| Hall | C3×C7⋊C9 ⋊ C22 |
| Sylow | C7 ⋊ C2×D3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | D3×AGL(1,𝔽7) |
| commutator, hypocenter | C3×C7 | C22×C9 |
| Fitting, socle, cosocle | C32×C7 | C22×C3 |
| prime radical | C3×D3×D7 | C3 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C3 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
63 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×D7⋊C9, D9×D7⋊C9, #72230062, … |
| maximal normal subgroup of | C2×D3×D7⋊C9, D3×C9×AGL(1,𝔽7), D3×#4430⋊C2, … |
| maximal quotient of | Dic3×D7⋊C9, D3×Dic7⋊C9, (D21⋊C4)⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 59);