Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C3×D3×#120 (order 1152) · Inn(G) = D3 · Out(G) = C2×C3×#120 |
| homology | Abelianization (H1) = C42×C7 · Schur multiplier (H2) = C4 |
| direct product | C4×C7×Dic3 |
| unambiguous semidirect product | C3⋊(C42×C7) |
| derived, Sylow | C3 ⋊ C42×C7 |
| Hall | C3×C7 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C7 | D3 |
| commutator, hypocenter | C3 | C42×C7 |
| Frattini | C22 | C2×D3×C7 |
| Fitting | C2×C3×C4×C7 | C2 |
| socle, prime radical | C22×C3×C7 | C22 |
| cosocle | C22×C3 | C22×C7 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C4×C7 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,3)(4,5,6,7,8,9,10)(15,16,17,18);(1,2,3)(11,12,13,14)168 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×C7⋊C8, #10620488, #10620512, … |
| maximal normal subgroup of | Dic3×C7⋊C8, #10620488, #10620512, … |
| maximal quotient of | C4×C7×C3⋊C8, C7×#803, C7×C8×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 81);