Finite Group DatabasebetaFinite Group Databasebeta| order | 14 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C3 (order 6) · Inn(G) = C1 · Out(G) = C2×C3 |
| holomorph | G ⋊ Aut(G) = C2×AGL(1,𝔽7) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C14 |
| abelian | C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 14 |
A minimal faithful permutation representation — 1 generator on 9 points:
(1,2)(3,4,5,6,7,8,9)14 conjugacy classes — view the full character table.
| center of | C7×D4, C7×Q8, C7×Dic3, … |
| derived subgroup of | Dic14, D28, C22⋊D7, … |
| Frattini subgroup of | Dic49, C4×C49, Dic98, … |
| Fitting subgroup of | Dic7, C2×D7, C2×ΓL(1,𝔽8), … |
| socle of | Dic7, C4×C7, C2×D7, … |
| maximal subgroup of | Dic7, C4×C7, C2×D7, … |
| maximal normal subgroup of | Dic7, C4×C7, C2×D7, … |
| maximal quotient of | C4×C7, C22×C7, D3×C7, … |
| outer automorphism group of | D29, C43⋊C3, C49⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(14, 2);