Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C7 · Schur multiplier (H2) = C26 |
| direct product | C23×D3×C7 |
| unambiguous semidirect product | C3⋊(C24×C7) |
| unambiguous semidirect product | (C2×C3)⋊(C23×C7) |
| Fitting, socle | C23×C3×C7 ⋊ C2 |
| derived, Sylow | C3 ⋊ C24×C7 |
| Hall | C3×C7 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C7 | D3 |
| commutator, cosocle, hypocenter | C3 | C24×C7 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C7 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C23×C7 | |
268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,4)(5,6)(8,9)(10,11,12,13,14,15,16);(5,6)(7,8,9);(3,4);(1,2)168 conjugacy classes — view the full character table.
| maximal subgroup of | C2×(C2×D3)⋊Dic7, D3×C22⋊Dic7, #10621130, … |
| maximal normal subgroup of | C2×(C2×D3)⋊Dic7, D3×C22⋊Dic7, #10621130, … |
| maximal quotient of | C22×C7×Dic6, C22×C4×D3×C7, C22×C7×D12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 226);