Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#26211 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C22×D4×C19 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C2×C4×D4×C19 ◁ G | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 51 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)(30,34)(31,32)(33,35)(36,37,40,41)(38,42,45,49)(39,46,48,43)(44,47,51,50);(20,22,23,27)(21,24,25,26)(29,32)(33,34)(36,38)(37,42)(39,47)(40,45)(41,49)(43,44)(46,51)(48,50);(28,30,31,35)(29,33,32,34)(36,39,40,48)(37,43,41,46)(38,44,45,51)(42,50,49,47);(21,25)(24,26)(29,32)(33,34)(36,37,40,41)(38,42,45,49)(39,43,48,46)(44,50,51,47)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 19199);