Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11140 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #396 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| prime radical | C24×C19 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C23×C4×C19 ◁ G | |
822 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)(21,23)(24,25,29,33,30,34,28,45)(26,35,39,47,38,51,37,31)(27,40,43,49,44,32,42,48)(36,52,54,55,41,50,53,46);(20,21,22,23)(24,26,28,37,30,38,29,39)(25,31,33,35,34,47,45,51)(27,41,42,54,44,36,43,53)(32,46,48,52,40,55,49,50);(24,27,30,44)(25,32,34,40)(26,36,38,41)(28,43,29,42)(31,46,47,55)(33,49,45,48)(35,50,51,52)(37,54,39,53)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4128);