Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C27 |
| direct product | C19×#21017 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #386 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | C2×#386 |
| prime radical | C19×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
1692 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)(20,21,25,31,26,30,24,29)(22,27,33,41,34,42,38,44)(23,35,36,45,37,28,39,43)(32,40,48,51,47,49,46,50);(20,22,25,38,26,34,24,33)(21,27,29,41,30,42,31,44)(23,32,39,48,37,47,36,46)(28,40,43,50,35,49,45,51);(20,23)(21,28)(22,32)(24,36)(25,39)(26,37)(27,40)(29,43)(30,35)(31,45)(33,46)(34,47)(38,48)(41,50)(42,49)(44,51);(20,24,26,25)(21,29,30,31)(22,33,34,38)(23,36,37,39)(27,41,42,44)(28,43,35,45)(32,46,47,48)(40,50,49,51)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14005);