Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C2×C42 |
| direct product | C19×#23351 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C19 | C24 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C23×C19 | C23×C4 |
| prime radical | C22×D4×C19 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C19 ◁ G | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
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)(21,24)(22,26)(28,29)(30,34)(31,33)(32,35)(36,37,41,45,40,42,39,44)(38,46,48,51,47,43,49,50);(20,21)(22,27)(23,24)(25,26)(30,35)(32,34)(37,42)(38,47)(44,45)(48,49);(21,24)(25,27)(28,30,31,35)(29,32,33,34)(36,38,40,47)(37,43,42,46)(39,49,41,48)(44,51,45,50);(20,22,23,26)(21,25,24,27)(29,33)(32,34)(36,39,40,41)(37,44,42,45)(38,48,47,49)(43,50,46,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 16339);