Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C22×C3×D3×D4 (order 576) · Inn(G) = C2×D3 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#235 |
| Fitting | C2×C3×C4×C7 ⋊ C2 |
| Hall | C3×C7 ⋊ #41 |
| Sylow | C3 ⋊ C7×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×D3 |
| commutator, cosocle | C2×C3 | C23×C7 |
| Frattini | C2 | C22×D3×C7 |
| Fitting | C2×C3×C4×C7 | C2 |
| socle | C2×C3×C7 | C23 |
| hypercenter | C2×C4×C7 | D3 |
| hypocenter | C3 | C7×#41 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3)(5,9)(8,11)(12,13,14,15,16,17,18);(4,10,7,6)(5,11,9,8);(2,3)(4,5)(6,8)(7,9)(10,11)126 conjugacy classes — view the full character table.
| maximal subgroup of | #10620513, #10620514, C7×#805, … |
| maximal normal subgroup of | #10620513, #10620514, C7×#805, … |
| maximal quotient of | C4×C7×Dic6, C7×#870, C7×#872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 187);