Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11182 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #397 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| prime radical | C24×C19 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C23×C4×C19 ◁ G | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25,29,35,30,33,28,34)(26,36,39,47,40,31,38,48)(27,41,43,50,45,51,44,32)(37,53,54,46,55,49,42,52);(20,21)(22,23)(24,26,30,40)(25,31,33,36)(27,42,45,54)(28,39,29,38)(32,49,50,53)(34,48,35,47)(37,43,55,44)(41,46,51,52);(21,23)(24,27)(25,32,33,50)(26,37)(28,43)(29,44)(30,45)(31,46,36,52)(34,41,35,51)(38,54)(39,42)(40,55)(47,49,48,53)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4170);