Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13286 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×D8 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×#102 |
| prime radical | C23×C19 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,26)(23,24)(25,27)(28,29,34,39,35,40,33,51)(30,41,45,53,46,58,44,56)(31,38,32,50,49,52,48,37)(36,42,54,59,55,47,57,43);(21,24)(25,26)(28,30,32,43,34,45,49,42,35,46,48,59,33,44,31,47)(29,36,52,58,51,57,50,53,40,55,38,41,39,54,37,56);(20,22,23,27)(21,25,24,26)(28,31,33,48,35,49,34,32)(29,37,39,38,40,50,51,52)(30,42,44,43,46,47,45,59)(36,53,54,58,55,56,57,41)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6274);