Finite Group DatabasebetaFinite Group Databasebeta| order | 1386 |
| isoclinism | isoclinic class of C3⋊AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C11 · Schur multiplier (H2) = C1 |
| Fitting, socle | C3×C7×C11 ⋊ C2×C3 |
| derived | C3×C7 ⋊ C2×C3×C11 |
| Hall | C3×C11×ΓL(1,𝔽8) ⋊ C2 |
| Hall | C7×C11 ⋊ C3×D3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C2×C11 |
| Sylow | C7 ⋊ C3×D3×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C11 | C3⋊AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C3×C7 | C2×C3×C11 |
| Frattini | C1 | G |
| Fitting, socle | C3×C7×C11 | C2×C3 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C11 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 231 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,14)(16,19,20,17,18,21);(12,13,14)(15,16,18,21,20,19,17)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1386, 18);