Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#32703 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | #2782 |
| prime radical | C19×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,29,26,31,24,30)(22,32,35,27,36,43,34,44)(23,37,41,28,39,47,40,46)(33,48,49,45,50,42,38,51);(20,22)(21,27)(23,38)(24,34)(25,35)(26,36)(28,45)(29,43)(30,32)(31,44)(33,41)(37,48)(39,49)(40,50)(42,47)(46,51);(20,23,24,40,26,39,25,41)(21,28,29,47,31,46,30,37)(22,33,35,49,36,50,34,38)(27,42,32,45,44,48,43,51);(22,34,36,35)(23,39)(27,43,44,32)(28,46)(33,49,50,38)(37,47)(40,41)(42,45,48,51)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 25691);