Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C23×AGL(1,𝔽7) (order 336) · Inn(G) = D7 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×C4×D7 |
| Fitting | C3×C4×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C2×C3×C4 |
| Hall | C3×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D7 |
| commutator, hypocenter | C7 | C2×C3×C4 |
| Frattini | C2 | C2×C3×D7 |
| Fitting | C3×C4×C7 | C2 |
| socle | C2×C3×C7 | C22 |
| prime radical | C2×C3×D7 | C2 |
| cosocle | C2×C7 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3)(4,5)(6,7)(12,13,14);(1,2,4,6,7,5,3)(8,11,10,9)60 conjugacy classes — view the full character table.
| maximal subgroup of | D7×C3⋊C8, #66557, C3×C8×D7, … |
| maximal normal subgroup of | D7×C3⋊C8, #66557, C3×C8×D7, … |
| maximal quotient of | C3×C8×D7, C3×#1131, C3×C4×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 25);