Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of D3×D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×D3×D7 |
| unambiguous semidirect product | (D3×C7)⋊C23 |
| unambiguous semidirect product | (C3×D7)⋊C23 |
| unambiguous semidirect product | D21⋊C23 |
| Fitting, socle | C22×C3×C7 ⋊ C22 |
| derived, Hall | C3×C7 ⋊ C24 |
| Sylow | C7 ⋊ C23×D3 |
| Sylow | C3 ⋊ C23×D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×D7 |
| commutator, cosocle, hypocenter | C3×C7 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C7 | C22 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C22 | |
268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2)(3,4)(5,6,7)(9,10)(11,12)(13,14);(6,7)(8,9,11,13,14,12,10);(3,4);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | D7×(C2×D3)⋊C4, D3×(C2×D7)⋊C4, #10621035, … |
| maximal normal subgroup of | D7×(C2×D3)⋊C4, D3×(C2×D7)⋊C4, #10621035, … |
| maximal quotient of | C2×D7×Dic6, C2×#66669, C2×#66670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 219);