Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #204 |
| automorphism | Aut(G) = C2×C3×D3×AGL(1,ℤ/8) (order 1152) · Inn(G) = D12 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×D24 |
| Fitting | C3×C7×C8 ⋊ C2 |
| Hall | C3×C7 ⋊ D8 |
| Sylow | C3 ⋊ C7×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | D12 |
| commutator, cosocle | C3×C4 | C22×C7 |
| Frattini | C4 | C2×D3×C7 |
| Fitting | C3×C7×C8 | C2 |
| socle | C2×C3×C7 | D4 |
| hypercenter | C7×C8 | D3 |
| hypocenter | C3 | C7×D8 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 ◁ 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:
(1,2,3,4,5,6,7)(8,9,10,11,12,13,14,15)(16,17,18);(9,15)(10,14)(11,13)(17,18)105 conjugacy classes — view the full character table.
| maximal subgroup of | #10620472, #10620474, #10620476, … |
| maximal normal subgroup of | #10620472, #10620474, #10620476, … |
| maximal quotient of | C7×D48, C7×#800, C7×Dic24, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 77);