Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11237 |
| 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 | |
622 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)(21,22)(24,25,29,33,30,34,28,44)(26,35,38,47,39,46,37,31)(27,40,42,49,43,55,41,32)(36,45,52,50,53,48,51,54);(20,21,23,22)(24,26,28,37,30,39,29,38)(25,31,33,35,34,47,44,46)(27,36,41,51,43,53,42,52)(32,48,40,54,49,45,55,50);(20,22,23,21)(24,27,28,41,30,43,29,42)(25,32,33,40,34,49,44,55)(26,36,37,51,39,53,38,52)(31,45,35,50,47,48,46,54)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4225);