Finite Group DatabasebetaFinite Group Databasebeta| order | 228 |
| isoclinism | isoclinic class of D19⋊C3 |
| automorphism | Aut(G) = C2×AGL(1,𝔽19) (order 684) · Inn(G) = D19⋊C3 · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×D19⋊C3 |
| unambiguous semidirect product | D19⋊(C2×C3) |
| unambiguous semidirect product | (C19⋊C3)⋊C22 |
| unambiguous semidirect product | (C2×D19)⋊C3 |
| unambiguous semidirect product | (C2×C19⋊C3)⋊C2 |
| Fitting, socle | C2×C19 ⋊ C2×C3 |
| derived, Sylow | C19 ⋊ C22×C3 |
| Hall | C2×D19 ⋊ C3 |
| Hall | C19⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D19⋊C3 |
| commutator, cosocle, hypocenter | C19 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C19 | C2×C3 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 38 |
A minimal faithful permutation representation — 2 generators on 21 points:
(4,8,9,5,6,10)(7,16,13,11,12,18)(14,19,21,20,17,15);(1,2)(3,4,7,13,19,16,20,6,10,15,21,9,8,14,12,17,18,11,5)18 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D19⋊C3, C4⋊(D19⋊C3), (C22⋊D19)⋊C3, … |
| maximal normal subgroup of | C4×D19⋊C3, C4⋊(D19⋊C3), (C22⋊D19)⋊C3, … |
| maximal quotient of | (C19⋊C3)⋊Q8, C4×D19⋊C3, C4⋊(D19⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(228, 7);