Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11250 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #389 |
| prime radical | C24×C19 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
618 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)(20,21)(22,23)(24,25,29,33,30,35,28,34)(26,36,39,47,40,50,38,31)(27,41,44,49,45,51,43,32)(37,53,54,48,55,46,42,52);(24,26,28,38,30,40,29,39)(25,31,33,36,35,47,34,50)(27,42,43,55,45,54,44,37)(32,48,41,46,49,52,51,53);(21,23)(24,27,28,43,30,45,29,44)(25,32,33,41,35,49,34,51)(26,37,38,42,40,55,39,54)(31,46,36,52,47,53,50,48)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4238);