Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9091 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11184 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #386 |
| 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 | |
818 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)(26,36)(27,41,45,50)(28,34)(29,33)(30,35)(31,38)(32,43,51,44)(37,46,54,55)(39,48)(40,47)(42,52,53,49);(20,21)(22,23)(24,26,30,40)(25,31,35,48)(27,42,45,53)(28,39,29,38)(32,49,51,52)(33,47,34,36)(37,43,54,44)(41,46,50,55);(21,23)(24,27,28,43,30,45,29,44)(25,32,33,50,35,51,34,41)(26,37,38,53,40,54,39,42)(31,46,36,52,48,55,47,49)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4172);