Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C23×C3×D4 (order 192) · Inn(G) = D4 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C1 |
| direct product | C3×C7×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C7 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C3×C7 |
| prime radical | C3×C7×D4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7 ◁ C3×C4×C7 ◁ G | |
40 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)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18);(12,14)(13,17)(16,18)147 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 112);