Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #213 |
| automorphism | Aut(G) = C22×C3×D3×D4 (order 576) · Inn(G) = #67 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#213 |
| Fitting | C3×C7×D4 ⋊ C2 |
| Hall | C3×C7 ⋊ D8 |
| Sylow | C3 ⋊ C7×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | #67 |
| commutator, cosocle | C3×C4 | C22×C7 |
| Frattini | C4 | C2×D3×C7 |
| Fitting | C3×C7×D4 | C2 |
| socle | C2×C3×C7 | D4 |
| hypercenter | C7×D4 | D3 |
| hypocenter | C3 | C7×D8 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 ◁ C7×D4 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3)(5,8)(6,11)(9,10)(12,13,14,15,16,17,18);(2,3)(4,5)(6,8)(7,10)(9,11)84 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#213, #10621040, #10621043, … |
| maximal normal subgroup of | D7×#213, #10621040, #10621043, … |
| maximal quotient of | C7×#807, C7×#809, C7×#826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 85);