Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×C4×D7 |
| Fitting | C2×C3×C4×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C3×C4 |
| Hall | C3×C7 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | D7 |
| commutator, hypocenter | C7 | C22×C3×C4 |
| Frattini | C2 | C22×C3×D7 |
| Fitting | C2×C3×C4×C7 | C2 |
| socle | C22×C3×C7 | C22 |
| prime radical | C22×C3×D7 | C2 |
| cosocle | C2×C7 | C23×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3×C4 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,4,5)(11,12)(13,14)(15,16);(6,7,8,9)(10,11,13,15,16,14,12);(1,2)120 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D7)⋊(C3⋊C8), C3×(C2×D7)⋊C8, C2×D7×C3⋊C8, … |
| maximal normal subgroup of | (C2×D7)⋊(C3⋊C8), C3×(C2×D7)⋊C8, C2×D7×C3⋊C8, … |
| maximal quotient of | C3×C4×Dic14, C3×C42×D7, C3×#6435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 175);