Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#13009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #481 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C24×C19 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C22×M16×C19 ◁ G | |
496 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)(21,23)(24,25,28,34,45,47,42,33,30,32,43,51,29,35,27,41)(26,36,39,31,46,49,37,53,38,54,55,48,40,52,44,50);(20,21,22,23)(24,26,29,46,30,38,45,40)(25,31,47,50,32,48,35,53)(27,39,43,37,42,55,28,44)(33,52,51,54,41,49,34,36);(25,32)(27,42)(28,43)(31,48)(35,47)(37,44)(39,55)(50,53)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5997);