Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C22×C42 |
| direct product | C19×#12581 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #99 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C22×D4 | C23×C19 |
| socle | C22×C19 | C2×#99 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C4×C19 ◁ C22×#41×C19 ◁ G | |
854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25)(26,30)(27,31,35,29)(28,32)(33,37,39,36)(34,38);(20,21,22,23)(25,29,32,31)(30,36,38,37);(24,26,28,34)(25,30,32,38)(27,33,35,39)(29,36,31,37)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5569);