Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11260 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #117 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #382 |
| prime radical | C24×C19 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C19×#117 ◁ G | |
590 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)(24,25,29,35,30,33,28,34)(26,36,38,48,39,31,43,46)(27,40,41,47,42,49,44,32)(37,51,53,50,54,55,52,45);(20,21)(22,23)(24,26,28,38,30,39,29,43)(25,31,34,46,33,36,35,48)(27,37,41,52,42,54,44,53)(32,45,40,55,47,50,49,51);(21,23)(24,27)(25,32,33,47)(26,37)(28,41)(29,44)(30,42)(31,45,36,50)(34,40,35,49)(38,52)(39,54)(43,53)(46,55,48,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4248);