Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×SD16⋊C16 |
| unambiguous semidirect product | SD16⋊(C16×C19) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C4 | C22×C16×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C22×C19 | C8×D4 |
| prime radical | C2×D4×C19 | C2×C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C4×C16×C19 ◁ G | |
232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 2432 |
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)(23,25)(26,27)(28,29,31,35,32,36,39,42,33,30,34,37,40,38,41,43);(21,24)(22,27)(25,26);(21,25)(24,26)(28,30,31,37,32,38,39,43,33,29,34,35,40,36,41,42)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5043);