Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#18016 |
| 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 | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C19×#431 ◁ G | |
600 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)(20,21,25,29)(22,27,31,34)(23,28,24,33)(26,32,35,30)(36,37,39,42,40,43,46,49)(38,41,45,48,44,47,50,51);(20,22)(21,26)(23,30)(24,32)(25,31)(27,33)(28,34)(29,35);(20,22)(21,27)(23,32)(24,30)(25,31)(26,33)(28,35)(29,34)(36,38,39,45,40,44,46,50)(37,41,42,48,43,47,49,51);(20,23,25,24)(21,28,29,33)(22,30,31,32)(26,34,35,27)(38,44)(41,47)(45,50)(48,51)1330 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 11004);