Finite Group DatabasebetaFinite Group Databasebeta| order | 228 |
| isoclinism | isoclinic class of D3×D19 |
| automorphism | Aut(G) = D3×AGL(1,𝔽19) (order 2052) · Inn(G) = G · Out(G) = C9 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | D3×D19 |
| unambiguous semidirect product | (D3×C19)⋊C2 |
| unambiguous semidirect product | (C3×D19)⋊C2 |
| unambiguous semidirect product | D57⋊C2 |
| Fitting, socle, derived, Hall | C3×C19 ⋊ C22 |
| Sylow | C19 ⋊ C2×D3 |
| Sylow | C3 ⋊ C2×D19 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C19 | C22 |
| LCS | G ▷ C3×C19 | |
| UCS | C1 | |
20 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);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(21,22)33 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×D19, D9×D19, D3×D19⋊C3, … |
| maximal normal subgroup of | C2×D3×D19, D3×D19⋊C3, C3×D3×D19, … |
| maximal quotient of | Dic3×D19, D3×Dic19, D57⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(228, 8);