Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#17953 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C23×C19 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C19×#431 ◁ G | |
558 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)(20,21,23,27,24,26,30,33)(22,25,29,32,28,31,34,35)(36,37,41,44,42,38,40,46)(39,43,49,50,47,45,48,51);(22,28)(25,31)(29,34)(32,35)(36,38,42,37)(39,43,47,45)(40,44,41,46)(48,51,49,50);(20,22,23,29,24,28,30,34)(21,25,27,32,26,31,33,35)(36,39,40,48,42,47,41,49)(37,43,46,51,38,45,44,50);(21,26)(25,31)(27,33)(32,35)(39,47)(43,45)(48,49)(50,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10941);