Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#11492 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×C22⋊C8 |
| prime radical | C19×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,25,23,24,26,27)(28,29,33,39,35,38,48,56,34,40,47,57,32,49,31,37)(30,41,44,55,43,36,42,52,45,54,58,51,50,53,46,59);(21,24)(22,26)(28,30)(29,36)(31,46)(32,50)(33,44)(34,45)(35,43)(37,55)(38,54)(39,52)(40,53)(41,49)(42,48)(47,58)(51,56)(57,59);(28,31,32,47,34,48,35,33)(29,37,49,57,40,56,38,39)(30,42,50,44,45,46,43,58)(36,51,41,52,53,55,54,59)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4480);