Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽7) (order 1344) · Inn(G) = D7 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×C7⋊C8 |
| derived, Sylow | C7 ⋊ C2×C3×C8 |
| Hall | C3×C7 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | D7 |
| commutator, hypocenter | C7 | C2×C3×C8 |
| Frattini | C4 | C2×C3×D7 |
| Fitting | C2×C3×C4×C7 | C2 |
| socle, prime radical | C22×C3×C7 | C4 |
| cosocle | C4×C7 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3×C4 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4,5)(7,8)(9,10)(11,12)(13,14,15,17,16,18,19,20);(1,2)(6,7,9,11,12,10,8)120 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×C7⋊C8, #10620486, #10620488, … |
| maximal normal subgroup of | Dic3×C7⋊C8, #10620486, #10620488, … |
| maximal quotient of | C3×C4×C7⋊C8, C3×C4⋊(C7⋊C8), C2×C3×C7⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 63);