Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#19566 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1216 |
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)(21,25)(23,26)(27,31)(28,32)(30,33)(34,35)(38,44)(39,46)(41,48)(43,50)(45,47)(49,51);(20,21,26,32,24,28,23,25)(22,27,33,35,29,34,30,31)(36,37)(38,41)(39,43)(40,42)(44,48)(45,49)(46,50)(47,51);(20,22)(21,27)(23,30)(24,29)(25,31)(26,33)(28,34)(32,35)(36,38,40,45)(37,41,42,49)(39,47,46,44)(43,51,50,48);(21,28)(22,29)(25,32)(30,33)(37,42)(41,49)(43,50)(48,51)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12554);