Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| automorphism | Aut(G) = C22×C3×D3×D4 (order 576) · Inn(G) = C22×D3 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | D3×C7×D4 |
| Fitting | C3×C7×D4 ⋊ C2 |
| Hall | C3×C7 ⋊ C2×D4 |
| Sylow | C3 ⋊ C2×C7×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C22×D3 |
| commutator, cosocle | C2×C3 | C23×C7 |
| Frattini | C2 | C22×D3×C7 |
| Fitting | C3×C7×D4 | C2 |
| socle | C2×C3×C7 | C23 |
| hypercenter | C7×D4 | D3 |
| hypocenter | C3 | C2×C7×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C7 ◁ C7×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:
(2,4)(5,6,7)(8,9,10,11,12,13,14);(1,2)(3,4);(6,7)105 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1141, D3×#1142, #10621056, … |
| maximal normal subgroup of | D3×#1141, D3×#1142, #10621056, … |
| maximal quotient of | C7×#878, D3×C7×C22⋊C4, C7×#67⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 188);