Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C23×C4 |
| direct product | C19×#21663 |
| 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×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 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)(28,29,33,40,35,41,34,39)(30,42,45,36,47,55,46,56)(31,48,32,38,51,57,50,37)(43,53,44,59,49,54,52,58);(21,24)(25,27)(28,30,34,46,35,47,33,45)(29,36,40,55,41,56,39,42)(31,49,50,44,51,43,32,52)(37,54,48,58,38,53,57,59);(20,22,23,26)(21,25,24,27)(28,31,33,32,35,51,34,50)(29,37,39,57,41,38,40,48)(30,43,45,44,47,49,46,52)(36,53,42,58,56,54,55,59);(28,32,33,51,35,50,34,31)(29,38,40,57,41,37,39,48)(30,44,45,49,47,52,46,43)(36,54,55,58,56,53,42,59)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14651);