Finite Group DatabasebetaFinite Group Databasebeta| order | 1386 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9×C11 · Schur multiplier (H2) = C1 |
| direct product | C2×C11×C7⋊C9 |
| unambiguous semidirect product | (C2×C7)⋊(C9×C11) |
| unambiguous semidirect product | (C2×C7×C11)⋊C9 |
| derived, Sylow | C7 ⋊ C2×C9×C11 |
| Hall | C2×C7×C11 ⋊ C9 |
| Hall | C7×C11 ⋊ C2×C9 |
| Hall | C2×C7 ⋊ C9×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C11 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C2×C9×C11 |
| Frattini | C3 | C2×C11×ΓL(1,𝔽8) |
| Fitting, socle, prime radical | C2×C3×C7×C11 | C3 |
| cosocle | C3×C7 | C2×C3×C11 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3×C11 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1386 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2)(3,4,5,6,7,8,9,10,11,12,13)(15,16,17)(18,20,19)(21,22,24,23,25,27,26,28,29);(14,15,16,18,17,19,20)330 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic11×C7⋊C9, C11×Dic7⋊C9, C11⋊(Dic7⋊C9), … |
| maximal normal subgroup of | Dic11×C7⋊C9, C11×Dic7⋊C9, C11⋊(Dic7⋊C9), … |
| maximal quotient of | C4×C11×C7⋊C9, C22×C11×C7⋊C9, C2×C7⋊C9×C121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1386, 4);