Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82×C19 · Schur multiplier (H2) = C2×C8 |
| direct product | C19×#11399 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C16×C19 | C22 |
| commutator | C2 | C2×C82×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C22×C19 | C2×C4×C8 |
| prime radical | C23×C19 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C16×C19 ◁ G | |
330 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)(28,29,31,35,32,36,41,47,33,37,42,48,43,49,53,57)(30,34,39,45,40,46,52,56,38,44,50,54,51,55,58,59);(20,21,22,24,23,25,26,27)(28,30)(29,34)(31,39)(32,40)(33,38)(35,45)(36,46)(37,44)(41,52)(42,50)(43,51)(47,56)(48,54)(49,55)(53,58)(57,59);(30,38)(34,44)(39,50)(40,51)(45,54)(46,55)(52,58)(56,59)3040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4387);