Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×C3×D3×#120 (order 1152) · Inn(G) = C2×D3 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C2×C7×#67 |
| Fitting | C23×C3×C7 ⋊ C2 |
| Hall | C3×C7 ⋊ C2×D4 |
| Sylow | C3 ⋊ C2×C7×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C7 | C2×D3 |
| commutator, cosocle | C2×C3 | C23×C7 |
| Frattini | C2 | C22×D3×C7 |
| Fitting | C23×C3×C7 | C2 |
| socle | C22×C3×C7 | C22 |
| hypercenter | C23×C7 | D3 |
| hypocenter | C3 | C2×C7×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C7 ◁ C23×C7 | |
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:
(3,4,5)(7,9)(10,11,12,13,14,15,16);(4,5)(6,7)(8,9);(1,2)126 conjugacy classes — view the full character table.
| maximal subgroup of | #10620531, C7×#806, #10621087, … |
| maximal normal subgroup of | #10620531, C7×#806, #10621087, … |
| maximal quotient of | C2×C7×Dic3⋊C4, C7×#924, C2×C7×(C2×D3)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 193);