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×#21659 |
| 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×C19×#97 ◁ G | |
722 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)(28,29,34,39,35,40,33,51)(30,41,45,53,46,57,44,36)(31,47,32,38,49,55,48,37)(42,54,50,59,58,52,43,56);(21,24)(25,27)(28,30,33,44,35,46,34,45)(29,36,39,41,40,53,51,57)(31,42,48,43,49,58,32,50)(37,52,47,56,38,54,55,59);(20,22,23,26)(21,25,24,27)(28,31,33,48,35,49,34,32)(29,37,39,47,40,38,51,55)(30,42,44,43,46,58,45,50)(36,52,41,56,53,54,57,59);(28,32,34,49,35,48,33,31)(29,38,39,55,40,37,51,47)(30,43,45,42,46,50,44,58)(36,52,41,56,53,54,57,59)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14647);