Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#10961 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | C4×C22⋊C4 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C8×C19 ◁ G | |
926 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)(20,21)(22,23)(24,25,28,33,29,32,41,45)(26,30,36,43,35,44,39,48)(27,31,38,46,37,47,40,49)(34,50,51,55,52,42,53,54);(21,23)(24,26)(25,30,32,44)(27,34)(28,39)(29,35)(31,42,47,50)(33,48,45,43)(36,41)(37,52)(38,53)(40,51)(46,55,49,54);(24,27,28,40,29,37,41,38)(25,31,45,46,32,47,33,49)(26,34,39,51,35,52,36,53)(30,42,43,55,44,50,48,54)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 3949);