Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11254 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #394 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C2×C19 | #1441 |
| prime radical | C24×C19 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,29,26,30,24,41)(22,31,34,44,35,43,33,27)(23,36,39,47,40,46,38,28)(32,42,48,45,49,50,37,51);(20,22,24,33,26,35,25,34)(21,27,29,31,30,44,41,43)(23,37,38,49,40,48,39,32)(28,45,36,50,47,51,46,42);(20,23,24,38,26,40,25,39)(21,28,29,36,30,47,41,46)(22,32,33,37,35,49,34,48)(27,42,31,45,44,50,43,51)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4242);