Finite Group DatabasebetaFinite Group Databasebeta| order | 1386 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)×D11 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| direct product | D11×C7⋊C9 |
| unambiguous semidirect product | C11⋊(C2×C7⋊C9) |
| unambiguous semidirect product | (C7×D11)⋊C9 |
| derived, Hall | C7×C11 ⋊ C2×C9 |
| Hall | C11×C7⋊C9 ⋊ C2 |
| Hall | C7×D11 ⋊ C9 |
| Sylow | C11 ⋊ C2×C7⋊C9 |
| Sylow | C7 ⋊ C9×D11 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | ΓL(1,𝔽8)×D11 |
| commutator, hypocenter | C7×C11 | C2×C9 |
| Fitting, socle, cosocle | C3×C7×C11 | C2×C3 |
| prime radical | C3×C7×D11 | C3 |
| LCS | G ▷ C7×C11 | |
| UCS | C1 ◁ C3 | |
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)(13,14,15)(16,18,17)(19,20,22,21,23,25,24,26,27);(2,11)(3,10)(4,9)(5,8)(6,7)(12,13,14,16,15,17,18)105 conjugacy classes — view the full character table.
| maximal subgroup of | D11×D7⋊C9, C2×D11×C7⋊C9, C7⋊C9×D121, … |
| maximal normal subgroup of | D11×D7⋊C9, C2×D11×C7⋊C9, C11×D11×C7⋊C9, … |
| maximal quotient of | Dic11×C7⋊C9, C2×D11×C7⋊C9, C7⋊C9×D121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1386, 1);