Finite Group DatabasebetaFinite Group Databasebeta| order | 1386 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×D7⋊C9 |
| unambiguous semidirect product | D7⋊(C9×C11) |
| unambiguous semidirect product | (C7⋊C9)⋊(C2×C11) |
| unambiguous semidirect product | (C11×D7)⋊C9 |
| derived, Sylow | C7 ⋊ C2×C9×C11 |
| Hall | C11×C7⋊C9 ⋊ C2 |
| Hall | C11×D7 ⋊ C9 |
| Hall | C7×C11 ⋊ C2×C9 |
| Hall | C7⋊C9 ⋊ C2×C11 |
| Hall | D7 ⋊ C9×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C11 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C2×C9×C11 |
| Frattini | C3 | C11×AGL(1,𝔽7) |
| Fitting, socle | C3×C7×C11 | C2×C3 |
| prime radical | C3×C11×D7 | C3 |
| cosocle | C3×C7 | C2×C3×C11 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C11 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 693 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,17,16,14,15,18)(19,23,27,21,26,22,24,20,25);(12,13,16,17,15,18,14)231 conjugacy classes — view the full character table.
| maximal subgroup of | D11×D7⋊C9, C2×C11×D7⋊C9, C121×D7⋊C9, … |
| maximal normal subgroup of | D11×D7⋊C9, C2×C11×D7⋊C9, C121×D7⋊C9, … |
| maximal quotient of | C11×Dic7⋊C9, C2×C11×D7⋊C9, C121×D7⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1386, 2);