Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33432 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C19 | C23 |
| commutator | C22 | C23×C8×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C22×C19 | C23×C8 |
| prime radical | C19×#525 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C19 ◁ G | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 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)(22,23)(24,25,27,32,28,33,37,45,29,31,38,43,39,44,49,53)(26,30,34,41,35,42,46,52,36,40,47,50,48,51,54,55);(21,23);(24,26,27,34,28,35,37,46,29,36,38,47,39,48,49,54)(25,30,32,41,33,42,45,52,31,40,43,50,44,51,53,55);(25,31)(30,40)(32,43)(33,44)(41,50)(42,51)(45,53)(52,55)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26420);