Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13172 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #466 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2432 |
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)(20,21,23,22)(25,31)(26,35)(27,29)(28,41)(32,45)(33,49)(34,40)(36,42)(37,51)(38,50)(39,52)(43,47)(48,55)(53,54);(20,22,23,21)(24,25)(26,32)(27,40)(28,33)(29,31)(30,34)(35,47)(36,46)(37,53)(38,43)(39,48)(41,44)(42,49)(45,51)(50,55)(52,54);(24,26,36,50,29,38,28,35,30,39,41,51,27,37,42,52)(25,32,46,55,31,43,33,47,34,48,44,45,40,53,49,54)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6160);