Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C27×C4 |
| direct product | C19×#20584 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C24 | C24×C19 |
| socle | C22×C19 | C22×C22⋊C4 |
| prime radical | C23×#41×C19 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C24×C4×C19 ◁ G | |
4006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23)(24,25)(26,32,34,29)(27,30)(28,31)(33,38,39,36)(35,37);(24,26)(25,29)(27,33)(28,34)(30,36)(31,32)(35,39)(37,38);(21,23);(24,27,28,35)(25,30,31,37)(26,33,34,39)(29,36,32,38)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13572);