Finite Group DatabasebetaFinite Group Databasebeta| order | 380 |
| isoclinism | isoclinic class of C19⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C19⋊AGL(1,𝔽5) |
| Fitting, socle, derived, Hall | C5×C19 ⋊ C4 |
| Sylow | C19 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ Dic19 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C19 | C4 |
| prime radical, cosocle | D5×C19 | C2 |
| LCS | G ▷ C5×C19 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 95 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(21,22,23,24);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(20,21,24,22,23)Polycyclic presentation:
2^2,5,19|a^2=b,c^a=c^3,d^a=d^18,c^b=c^441 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(380, 6);