Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#17978 |
| 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 | C19×#466 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
606 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)(22,29)(23,27)(24,32)(25,28)(26,30)(31,34)(33,35)(36,37,41,44,42,38,40,45)(39,46,48,51,49,43,47,50);(20,22)(21,26)(23,30)(24,33)(25,31)(27,34)(28,35)(29,32)(36,38)(37,42)(39,46)(40,44)(41,45)(43,49)(47,50)(48,51);(36,39,41,48,42,49,40,47)(37,43,44,50,38,46,45,51);(20,23,25,24)(21,27,28,32)(22,30,31,33)(26,34,35,29)(36,40,42,41)(37,44,38,45)(39,47,49,48)(43,50,46,51)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10966);