Finite Group DatabasebetaFinite Group Databasebeta| order | 228 |
| isoclinism | isoclinic class of C19⋊C3 |
| automorphism | Aut(G) = C2×AGL(1,𝔽19) (order 684) · Inn(G) = C19⋊C3 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C4×C19⋊C3 |
| unambiguous semidirect product | (C4×C19)⋊C3 |
| Fitting, Hall | C4×C19 ⋊ C3 |
| derived, Sylow | C19 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C19⋊C3 |
| commutator, hypocenter | C19 | C3×C4 |
| Frattini | C2 | C2×C19⋊C3 |
| Fitting | C4×C19 | C3 |
| socle, cosocle | C2×C19 | C2×C3 |
| prime radical | C2×C19⋊C3 | C2 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C4 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 76 |
A minimal faithful permutation representation — 2 generators on 23 points:
(2,3,5)(4,7,8)(6,10,9)(11,15,13)(12,16,14)(17,19,18)(20,21,22,23);(1,2,4,8,13,18,15,3,6,11,16,5,9,14,7,12,17,19,10)36 conjugacy classes — view the full character table.
| maximal subgroup of | (C19⋊C3)⋊C8, C8×C19⋊C3, (C19⋊C3)⋊Q8, … |
| maximal normal subgroup of | (C19⋊C3)⋊C8, C8×C19⋊C3, (C19⋊C3)⋊Q8, … |
| maximal quotient of | C8×C19⋊C3, C2×C4×C19⋊C3, C4×#4294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(228, 2);