Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#19515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
576 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)(20,21,24,28)(22,25,29,33)(23,26,31,27)(30,32,35,34)(37,41)(38,45)(39,42)(44,49)(46,47)(48,50);(20,22)(21,25)(23,30)(24,29)(26,32)(27,34)(28,33)(31,35)(36,37,42,49,40,44,39,41)(38,43,45,50,46,51,47,48);(21,26)(23,31)(25,32)(27,28)(30,35)(33,34)(36,38)(37,43)(39,47)(40,46)(41,48)(42,45)(44,51)(49,50);(22,29)(25,33)(30,35)(32,34)(37,44)(38,46)(41,49)(45,47)703 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12503);