Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#21735 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C23×C19 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
662 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)(22,27)(23,25)(24,26)(28,29,34,39,35,40,33,50)(30,36,44,52,45,53,43,58)(31,46,32,38,49,55,48,37)(41,56,57,54,47,59,42,51);(20,22,24,23)(21,25,26,27)(28,30)(29,36)(31,47)(32,42)(33,43)(34,44)(35,45)(37,54)(38,51)(39,52)(40,53)(41,49)(46,59)(48,57)(50,58)(55,56);(28,31,33,48,35,49,34,32)(29,37,39,46,40,38,50,55)(30,41,43,42,45,47,44,57)(36,51,52,56,53,54,58,59);(20,23,24,22)(21,25,26,27)(28,32,34,49,35,48,33,31)(29,38,39,55,40,37,50,46)(30,42,44,41,45,57,43,47)(36,51,52,56,53,54,58,59)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14723);