Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#8248 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | C4×C22⋊C4 |
| prime radical | C19×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,28)(23,31)(24,27)(25,30)(26,32)(29,34)(33,35)(36,37,41,44,42,38,40,46)(39,43,48,50,49,45,47,51);(20,22)(21,25,27,33)(23,30)(24,29)(26,28,31,34)(32,35)(36,38,42,37)(39,43,49,45)(40,44,41,46)(47,51,48,50);(22,29)(25,33)(28,34)(30,35)(36,39,41,48,42,49,40,47)(37,43,44,50,38,45,46,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1236);