Finite Group DatabasebetaFinite Group Databasebeta| order | 19350 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C3⋊((C5⋊D5)⋊(C43⋊C3)) |
| unambiguous semidirect product | (C3⋊(C5⋊D5))⋊(C43⋊C3) |
| Fitting, socle, derived | C3×C52×C43 ⋊ C2×C3 |
| Hall | C3×C52⋊(C43⋊C3) ⋊ C2 |
| Hall | C52×C43 ⋊ C3×D3 |
| Sylow | C43 ⋊ C3⋊((C5⋊D5)⋊C3) |
| Sylow | C52 ⋊ D3×C43⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C52×C43 | C2×C3 |
| LCS | G ▷ C3×C52×C43 | |
| UCS | C1 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 1935 |
A minimal faithful permutation representation — 2 generators on 61 points:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,33)(18,26,31)(21,29,28)(23,32,25)(27,36,40)(30,37,39)(34,41,35)(38,42,43)(44,45,47)(46,56,52,50,49,53)(48,58,57,54,55,51)(60,61);(1,2,4,8,15,20,5,9,16,25,35,32,12,19,28,24,34,41,22,31,39,36,14,23,33,40,43,42,26,7,13,21,30,38,37,17,3,6,11,18,27,29,10)(44,46,51,57,50)(45,48,49,56,54)(47,52,58,55,53)(59,60,61)Polycyclic presentation:
2,3^2,5^2,43|c^a=c^2,d^a=d^4,e^a=e^4,d^b=de^3,e^b=d^4e^3,f^b=f^36567 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19350, 91);