Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11241 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #380 |
| prime radical | C24×C19 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
638 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)(24,25)(26,34)(27,31,39,45)(28,40)(29,33)(30,36)(32,41)(35,46,49,55)(37,47)(38,51,52,44)(42,53,54,48)(43,50);(20,21)(22,23)(24,26,28,36,29,37,41,50)(25,30,32,34,33,43,40,47)(27,35,38,48,39,49,52,53)(31,42,44,46,45,54,51,55);(21,23)(24,27,28,38,29,39,41,52)(25,31,32,44,33,45,40,51)(26,35,36,48,37,49,50,53)(30,42,34,46,43,54,47,55)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4229);