Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = AGL(1,𝔽7) · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×ΓL(1,𝔽8)⋊C8 |
| unambiguous semidirect product | (C2×ΓL(1,𝔽8))⋊C8 |
| unambiguous semidirect product | (C2×C7⋊C8)⋊C3 |
| derived, Sylow | C7 ⋊ C2×C3×C8 |
| Hall | C2×C7⋊C8 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C2×C8 |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 17 points:
(4,8,7,5,6,9)(10,11,12,14,13,15,16,17);(1,2)(3,4,7,8,6,9,5)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 12);