Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C24×C3×D3 (order 288) · Inn(G) = D3 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C2 |
| direct product | D3×C7×C8 |
| Fitting | C3×C7×C8 ⋊ C2 |
| derived, Sylow | C3 ⋊ C2×C7×C8 |
| Hall | C3×C7 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C8 | D3 |
| commutator, hypocenter | C3 | C2×C7×C8 |
| Frattini | C4 | C2×D3×C7 |
| Fitting | C3×C7×C8 | C2 |
| socle | C2×C3×C7 | C2×C4 |
| prime radical | C2×D3×C7 | C4 |
| cosocle | C3×C4 | C22×C7 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C7×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)(12,13,14,15,16,17,18);(1,2,3)(4,11,10,9,7,8,6,5)168 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C7⋊C16, #10620470, D3×C7×C16, … |
| maximal normal subgroup of | D3×C7⋊C16, #10620470, D3×C7×C16, … |
| maximal quotient of | D3×C7×C16, C7×#798, C7×C8×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 74);