Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#11697 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #99 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×C4⋊C4 | C23×C19 |
| socle | C22×C19 | C2×#99 |
| prime radical | C19×#392 | C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C4×C19 ◁ C2×C19×#117 ◁ G | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,24)(23,26)(25,27)(28,29,31,35,32,33,38,40)(30,34,36,41,37,39,42,43);(21,24)(22,27)(25,26)(29,33)(34,39);(28,30,32,37)(29,34,33,39)(31,36,38,42)(35,41,40,43)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4685);