Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = D7 · Out(G) = C2×C3×D4 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×Dic7 |
| derived, Sylow | C7 ⋊ C2×C3×C4 |
| Hall | C3×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D7 |
| commutator, hypocenter | C7 | C2×C3×C4 |
| Frattini | C2 | C2×C3×D7 |
| Fitting, socle, prime radical | C22×C3×C7 | C2 |
| cosocle | C2×C7 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C3 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,4,5)(6,7,8,9)(11,12)(13,14)(15,16);(1,2)(10,11,13,15,16,14,12)60 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×Dic7, (C2×D3)⋊Dic7, #66575, … |
| maximal normal subgroup of | Dic3×Dic7, (C2×D3)⋊Dic7, #66575, … |
| maximal quotient of | C2×C3×C7⋊C8, C3×#1136, C3×C4×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 27);