Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×C3×S4 (order 288) · Inn(G) = C22 · Out(G) = C22×C3×D3 |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C1 |
| direct product | C3×C7×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C3×C7 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C3×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7 ◁ G | |
24 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)(15,18,17,16);(11,15,13,17)(12,16,14,18)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 41);