Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11086 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #124 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #397 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
866 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)(26,31,39,47,37,48,38,46)(27,40,44,50,42,32,43,51)(36,45,53,54,52,55,41,49);(20,21)(22,23)(24,26)(25,31)(27,41,42,53)(28,38)(29,39)(30,37)(32,49,40,54)(33,48)(34,46)(35,47)(36,43,52,44)(45,50,55,51);(21,23)(24,27)(25,32)(26,36)(28,44)(29,43)(30,42)(31,45)(33,40)(34,51)(35,50)(37,52)(38,53)(39,41)(46,54)(47,49)(48,55)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4074);