Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#12994 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C2×D8 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C24×C19 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C23×C4×C19 ◁ C23×C8×C19 ◁ G | |
530 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)(21,23)(24,25)(26,27)(28,29,33,38,48,57,46,55,34,39,32,37,31,36,47,56)(30,40,43,53,41,59,49,35,45,52,42,51,44,58,50,54);(20,21,22,23)(28,30)(29,35)(31,41)(32,49)(33,50)(34,45)(36,51)(37,58)(38,59)(39,54)(40,55)(42,46)(43,47)(44,48)(52,56)(53,57);(25,27)(28,31,34,48)(29,36,39,57)(30,41,45,44)(32,46,33,47)(35,51,54,53)(37,55,38,56)(40,59,52,58)(42,50,43,49)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5982);