Finite Group DatabasebetaFinite Group Databasebeta| order | 285 |
| isoclinism | isoclinic class of C19⋊C3 |
| automorphism | Aut(G) = C4×AGL(1,𝔽19) (order 1368) · Inn(G) = C19⋊C3 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×C19⋊C3 |
| unambiguous semidirect product | C19⋊(C3×C5) |
| unambiguous semidirect product | (C5×C19)⋊C3 |
| Fitting, socle, Hall | C5×C19 ⋊ C3 |
| derived, Sylow | C19 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | C19⋊C3 |
| commutator, cosocle, hypocenter | C19 | C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C5×C19 | C3 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C5 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 95 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,4,5)(7,8,10)(9,12,13)(11,15,14)(16,20,18)(17,21,19)(22,24,23);(6,7,9,13,18,23,20,8,11,16,21,10,14,19,12,17,22,24,15)45 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D19⋊C3, D5×C19⋊C3, C5⋊(D19⋊C3), … |
| maximal normal subgroup of | C5×D19⋊C3, D5×C19⋊C3, C5⋊(D19⋊C3), … |
| maximal quotient of | C2×C5×C19⋊C3, C5×#4294, C3×C5×C19⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(285, 1);