Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×C3×C4×D3 (order 288) · Inn(G) = D3 · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C7×C16 · Schur multiplier (H2) = C1 |
| direct product | C7×C3⋊C16 |
| unambiguous semidirect product | C3⋊(C7×C16) |
| derived, Sylow | C3 ⋊ C7×C16 |
| Hall | C3×C7 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C8 | D3 |
| commutator, hypocenter | C3 | C7×C16 |
| Frattini | C8 | D3×C7 |
| Fitting | C3×C7×C8 | C2 |
| socle, prime radical | C2×C3×C7 | C8 |
| cosocle | C3×C8 | C2×C7 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C7×C8 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 2 generators on 26 points:
(2,3)(4,15,18,10,14,9,13,5,8,19,12,17,7,16,6,11)(20,21,22,23,24,25,26);(1,2,3)168 conjugacy classes — view the full character table.
| maximal subgroup of | D7×C3⋊C16, D21⋊C16, #10620469, … |
| maximal normal subgroup of | D7×C3⋊C16, D21⋊C16, #10620469, … |
| maximal quotient of | C7×C3⋊C32, C2×C7×C3⋊C16, C7×C9⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 3);