Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#21184 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C2×#121 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | C2×#389 |
| prime radical | C24×C19 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
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)(20,21,25,29,26,30,24,40)(22,31,33,27,35,42,34,43)(23,36,39,44,37,50,38,28)(32,41,47,49,46,51,48,45);(20,22,25,33,26,35,24,34)(21,27,29,42,30,43,40,31)(23,32,39,47,37,46,38,48)(28,41,36,49,44,51,50,45);(20,23,24,38,26,37,25,39)(21,28,29,36,30,44,40,50)(22,32,34,48,35,46,33,47)(27,41,42,49,43,51,31,45);(23,37)(28,44)(32,46)(36,50)(38,39)(41,51)(45,49)(47,48)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14172);