Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = C3×D3×AGL(1,𝔽7) (order 756) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C2×C3×D3 |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C3 |
| direct product | C2×C9×ΓL(1,𝔽8) |
| Fitting | C2×C7×C9 ⋊ C3 |
| derived, Sylow | C7 ⋊ C2×C3×C9 |
| Hall | C2×C7 ⋊ C3×C9 |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(4,5,6)(7,9,8);(3,4,5,7,6,8,9)(10,14,18,12,17,13,15,11,16)90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 23);