Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11218 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #386 |
| 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)(21,22)(24,25,29,33,30,35,28,34)(26,31,38,46,39,47,37,51)(27,40,42,32,43,49,44,50)(36,48,52,45,53,55,41,54);(20,21,23,22)(24,26)(25,31)(27,41)(28,37)(29,38)(30,39)(32,48)(33,46)(34,51)(35,47)(36,42)(40,54)(43,52)(44,53)(45,49)(50,55);(20,22,23,21)(24,27)(25,32)(26,36)(28,44)(29,42)(30,43)(31,45)(33,49)(34,40)(35,50)(37,41)(38,52)(39,53)(46,55)(47,54)(48,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4206);