Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#8410 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | #379 |
| prime radical | C24×C19 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ G | |
692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,27,25,22,23,26)(28,29,32,37,33,36,47,51)(30,38,41,52,40,34,45,49)(31,42,44,53,43,35,46,50)(39,54,56,59,55,48,57,58);(20,22,25,21)(23,26,24,27)(28,30)(29,34)(31,39)(32,45)(33,40)(35,48)(36,38)(37,52)(41,47)(42,54)(43,55)(44,57)(46,56)(49,51)(50,59)(53,58);(28,31)(29,35)(30,39)(32,46)(33,43)(34,48)(36,42)(37,53)(38,54)(40,55)(41,57)(44,47)(45,56)(49,59)(50,51)(52,58)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1398);