Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽7) (order 1344) · Inn(G) = C22×D7 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×D4×D7 |
| Fitting | C3×C7×D4 ⋊ C2 |
| Hall | C3×C7 ⋊ C2×D4 |
| Sylow | C7 ⋊ C2×C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×D7 |
| commutator, cosocle | C2×C7 | C23×C3 |
| Frattini | C2 | C22×C3×D7 |
| Fitting | C3×C7×D4 | C2 |
| socle | C2×C3×C7 | C23 |
| hypercenter | C3×D4 | D7 |
| hypocenter | C7 | C2×C3×D4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 3 generators on 14 points:
(2,3)(4,5)(6,7)(12,13,14);(1,2,4,6,7,5,3)(9,11);(8,9)(10,11)75 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#213, D7×#214, #10621051, … |
| maximal normal subgroup of | D7×#213, D7×#214, #10621051, … |
| maximal quotient of | C3×#6441, C3×D7×C22⋊C4, C3×(C22⋊D7)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 178);