Finite Group DatabasebetaFinite Group Databasebeta| order | 19950 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C25⋊#11716328 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C19×C25 | C2×C3 |
| Frattini | C5 | C5⋊#11716328 |
| socle | C5×C7×C19 | C3×D5 |
| LCS | G ▷ C7×C19×C25 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 3325 |
A minimal faithful permutation representation — 2 generators on 51 points:
(2,6,7,3,4,8)(5,13,14,9,10,18)(11,19,17,16,15,12)(21,24,25,22,23,26)(28,30)(29,33)(31,36)(32,39)(34,42)(35,41)(37,38)(40,46)(43,47)(44,48)(45,50)(49,51);(1,2,5,10,15,18,17,7,6,11,16,4,8,12,14,19,13,9,3)(20,21,23,26,25,24,22)(27,41,51,44,30,33,47,45,36,39,42,40,38,37,46,34,32,31,50,43,29,28,48,49,35)Polycyclic presentation:
2,3,5,7,19,5|c^5=f^4,c^a=c^4f,d^a=d^6,e^a=e^18,f^a=f^4,d^b=d^2,e^b=e^11592 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19950, 48);