Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | Aut(G) = #120×AGL(1,𝔽7) (order 1344) · Inn(G) = AGL(1,𝔽7) · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C4×AGL(1,𝔽7) |
| unambiguous semidirect product | D7⋊(C2×C3×C4) |
| unambiguous semidirect product | (C2×C4×D7)⋊C3 |
| Fitting | C2×C4×C7 ⋊ C2×C3 |
| derived, Sylow | C7 ⋊ C22×C3×C4 |
| Hall | C2×C4×D7 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C22×C4 |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 13 points:
(8,11,12,9,10,13);(3,4,5,6)(7,8,10,13,12,11,9);(1,2)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 122);