Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of D21 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C3⋊(C7⋊C8) |
| unambiguous semidirect product | (C2×C3)⋊(C7⋊C8) |
| unambiguous semidirect product | (C2×C7)⋊(C3⋊C8) |
| derived, Hall | C3×C7 ⋊ C2×C8 |
| Sylow | C7 ⋊ C2×C3⋊C8 |
| Sylow | C3 ⋊ C2×C7⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D21 |
| commutator, hypocenter | C3×C7 | C2×C8 |
| Frattini | C4 | C2×D21 |
| Fitting | C2×C3×C4×C7 | C2 |
| socle, prime radical | C22×C3×C7 | C4 |
| cosocle | C3×C4×C7 | C22 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×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)(4,5)(7,8)(9,10)(11,12)(13,14,15,17,16,18,19,20);(1,2)(3,4,5)(6,7,9,11,12,10,8)96 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊C8×Dic7, Dic3×C7⋊C8, #10620487, … |
| maximal normal subgroup of | C3⋊C8×Dic7, Dic3×C7⋊C8, #10620487, … |
| maximal quotient of | C4×C3⋊(C7⋊C8), C4⋊(C3⋊(C7⋊C8)), C2×C3⋊(C7⋊C16), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 95);