Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#21971 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C22×C4⋊C4 |
| prime radical | C24×C19 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C19×#117 ◁ 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)(22,26)(23,25)(24,27)(28,29,33,37,34,38,32,47)(30,35,42,50,40,49,41,56)(31,43,45,51,46,57,44,36)(39,52,55,58,53,59,54,48);(20,22,23,27)(21,24,25,26)(28,30)(29,35)(31,39)(32,41)(33,42)(34,40)(36,48)(37,50)(38,49)(43,52)(44,54)(45,55)(46,53)(47,56)(51,58)(57,59);(22,27)(24,26)(28,31,32,44,34,46,33,45)(29,36,37,43,38,51,47,57)(30,39,41,54,40,53,42,55)(35,48,50,52,49,58,56,59);(21,25)(24,26)(30,40)(35,49)(39,53)(41,42)(48,58)(50,56)(52,59)(54,55)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14959);