Finite Group DatabasebetaFinite Group Databasebeta| order | 114 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×D3×C9 (order 108) · Inn(G) = D3 · Out(G) = C2×C9 |
| holomorph | G ⋊ Aut(G) = D32×AGL(1,𝔽19) |
| homology | Abelianization (H1) = C2×C19 · Schur multiplier (H2) = C1 |
| direct product | D3×C19 |
| unambiguous semidirect product | C3⋊(C2×C19) |
| Fitting, socle, Hall | C3×C19 ⋊ C2 |
| derived, Sylow | C3 ⋊ C2×C19 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C19 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C19 |
| Frattini | C1 | G |
| Fitting, socle | C3×C19 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C19 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 57 |
A minimal faithful permutation representation — 2 generators on 22 points:
(2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22);(1,2,3)57 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D19, C2×D3×C19, D9×C19, … |
| maximal normal subgroup of | D3×D19, C2×D3×C19, D3×C19⋊C3, … |
| maximal quotient of | Dic3×C19, C2×D3×C19, D9×C19, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(114, 3);