Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11306 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #124 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #400 |
| prime radical | C24×C19 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
530 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)(24,25,30,33,29,34,28,35)(26,36,40,46,39,47,38,31)(27,32,43,49,44,50,42,51)(37,48,41,54,53,55,52,45);(20,21)(22,23)(24,26,30,40,29,39,28,38)(25,31,35,47,34,46,33,36)(27,41,42,37,44,52,43,53)(32,48,49,54,50,55,51,45);(21,23)(24,27,28,42,29,44,30,43)(25,32,33,49,34,50,35,51)(26,37,38,52,39,53,40,41)(31,45,36,48,46,54,47,55)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4294);