Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#21709 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C19×#3150 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
1006 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)(21,23)(24,25,29,33,30,34,28,44)(26,35,39,45,36,50,38,31)(27,32,42,48,43,49,41,55)(37,51,53,47,40,54,52,46);(24,26,28,38,30,36,29,39)(25,31,33,35,34,45,44,50)(27,40,41,53,43,37,42,52)(32,47,48,54,49,46,55,51);(20,21,22,23)(26,36)(31,45)(35,50)(37,40)(38,39)(46,47)(51,54)(52,53);(24,27,29,42,30,43,28,41)(25,32,33,48,34,49,44,55)(26,37,39,53,36,40,38,52)(31,46,35,51,45,47,50,54)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14697);