Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C28 |
| direct product | C19×#20612 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C2×#120 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | C2×#379 |
| prime radical | C2×C19×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C23×C4×C19 ◁ G | |
3640 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)(20,21,24,29,25,28,37,42)(22,30,33,43,32,26,36,39)(23,27,35,41,34,40,47,50)(31,38,46,49,45,44,48,51);(20,22)(21,26)(23,31)(24,36)(25,32)(27,38)(28,30)(29,43)(33,37)(34,45)(35,48)(39,42)(40,44)(41,51)(46,47)(49,50);(20,23)(21,27)(22,31)(24,35)(25,34)(26,38)(28,40)(29,41)(30,44)(32,45)(33,46)(36,48)(37,47)(39,49)(42,50)(43,51);(21,28)(26,30)(27,40)(29,42)(38,44)(39,43)(41,50)(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, 13600);