Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×C3×D3×D4 (order 576) · Inn(G) = D3 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×C3⋊C8 |
| unambiguous semidirect product | (C2×C3)⋊(C7×C8) |
| derived, Sylow | C3 ⋊ C2×C7×C8 |
| Hall | C3×C7 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C7 | D3 |
| commutator, hypocenter | C3 | C2×C7×C8 |
| Frattini | C4 | C2×D3×C7 |
| Fitting | C2×C3×C4×C7 | C2 |
| socle, prime radical | C22×C3×C7 | C4 |
| cosocle | C3×C4 | C22×C7 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C4×C7 | |
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)(4,5)(6,7,8,9,10,11,12)(13,14,15,17,16,18,19,20);(1,2)(3,4,5)168 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊C8×Dic7, #10620486, #10620487, … |
| maximal normal subgroup of | C3⋊C8×Dic7, #10620486, #10620487, … |
| maximal quotient of | C4×C7×C3⋊C8, C7×C4⋊(C3⋊C8), C2×C7×C3⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 79);