Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of D7×D27 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | D7×D27 |
| unambiguous semidirect product | (C7×D27)⋊C2 |
| unambiguous semidirect product | (D7×C27)⋊C2 |
| unambiguous semidirect product | D189⋊C2 |
| Fitting, derived, Hall | C7×C27 ⋊ C22 |
| Sylow | C27 ⋊ C2×D7 |
| Sylow | C7 ⋊ C2×D27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C27 | C22 |
| Frattini | C9 | D3×D7 |
| socle | C3×C7 | C2×D9 |
| LCS | G ▷ C7×C27 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 189 |
75 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D7×D27, D7×D27⋊C3, AGL(1,𝔽7)×D27, … |
| maximal normal subgroup of | C2×D7×D27, D7×D27⋊C3, AGL(1,𝔽7)×D27, … |
| maximal quotient of | #13094115, Dic7×D27, D189⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 8);