Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#18086 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C23×C19 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C19×#431 ◁ G | |
572 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)(20,21,25,28,26,22,24,30)(23,27,32,34,33,29,31,35)(36,37,41,44,40,43,49,51)(38,45,47,48,46,42,50,39);(20,22,26,21)(23,29,33,27)(24,28,25,30)(31,34,32,35)(36,38)(37,42)(39,44)(40,46)(41,47)(43,45)(48,51)(49,50);(20,23,24,31,26,33,25,32)(21,27,30,35,22,29,28,34)(36,39,41,45,40,48,49,42)(37,38,44,47,43,46,51,50);(21,22)(27,29)(28,30)(34,35)(38,46)(39,48)(42,45)(47,50)1330 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 11074);