Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#8917 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C42 | C22×C4×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | #385 |
| prime radical | C19×#121 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,29,26,30,24,32)(22,31,28,27,34,35,33,23)(36,37,40,43)(38,44,45,51)(39,47,48,42)(41,49,50,46);(20,22,25,28,26,34,24,33)(21,27,29,35,30,23,32,31)(39,48)(42,47)(44,51)(46,49);(20,23)(21,28)(22,32)(24,35)(25,31)(26,27)(29,34)(30,33)(36,38)(37,41)(39,49)(40,45)(42,51)(43,50)(44,47)(46,48)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1905);