Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×S4×AGL(1,𝔽7) (order 2016) · Inn(G) = D7 · Out(G) = C2×C3×S4 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C22×C3×D7 |
| Fitting, socle | C22×C3×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C23×C3 |
| Hall | C3×C7 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D7 |
| commutator, cosocle, hypocenter | C7 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C3 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(3,4)(5,6,7)(9,10)(11,12)(13,14);(3,4)(8,9,11,13,14,12,10);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D7)⋊Dic3, C3×(C2×D7)⋊C4, C2×Dic3×D7, … |
| maximal normal subgroup of | (C2×D7)⋊Dic3, C3×(C2×D7)⋊C4, C2×Dic3×D7, … |
| maximal quotient of | C2×C3×Dic14, C2×C3×C4×D7, C2×C3×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 54);