Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×C3×#120 (order 384) · Inn(G) = C22 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C2×C3×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C3×C7×C4⋊C4 |
| unambiguous semidirect product | C4⋊(C3×C4×C7) |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3×C7 | C22 |
| commutator | C2 | C2×C3×C4×C7 |
| Frattini, cosocle | C22 | C22×C3×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4)(6,8)(9,10,11,12,13,14,15)(16,17,18);(5,6,7,8)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 108);