Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C24×AGL(1,𝔽7) (order 672) · Inn(G) = D7 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×C8×D7 |
| Fitting | C3×C7×C8 ⋊ C2 |
| derived, Sylow | C7 ⋊ C2×C3×C8 |
| Hall | C3×C7 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | D7 |
| commutator, hypocenter | C7 | C2×C3×C8 |
| Frattini | C4 | C2×C3×D7 |
| Fitting | C3×C7×C8 | C2 |
| socle | C2×C3×C7 | C2×C4 |
| prime radical | C2×C3×D7 | C4 |
| cosocle | C4×C7 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C8 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,3)(4,5)(6,7)(16,17,18);(1,2,4,6,7,5,3)(8,12,14,9,11,15,10,13)120 conjugacy classes — view the full character table.
| maximal subgroup of | D7×C3⋊C16, #10620469, C3×D7×C16, … |
| maximal normal subgroup of | D7×C3⋊C16, #10620469, C3×D7×C16, … |
| maximal quotient of | C3×D7×C16, C3×#6372, C3×C8×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 58);