Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11130 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #392 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| prime radical | C24×C19 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C23×C4×C19 ◁ G | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23)(24,25)(26,35)(27,32)(28,44)(29,33)(30,34)(31,37)(36,47)(38,52)(39,46)(40,55)(41,50)(42,48)(43,49)(45,53)(51,54);(24,26,28,37,30,38,29,39)(25,31,33,35,34,46,44,52)(27,40,41,36,43,53,42,54)(32,47,48,55,49,51,50,45);(24,27,30,43)(25,32,34,49)(26,36,38,54)(28,42,29,41)(31,45,46,55)(33,50,44,48)(35,51,52,47)(37,40,39,53)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4118);