Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#7193 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #99 |
| commutator | C2×C4 | C4×C8×C19 |
| Frattini, cosocle | C22×M16 | C22×C19 |
| socle | C22×C19 | #324 |
| prime radical | C24×C19 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C4×C19 ◁ C22×M16×C19 ◁ G | |
342 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25,27,32,29,30,35,37,28,33,36,39,26,31,34,38);(20,21,22,23)(25,30,33,31)(32,38,39,37)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 181);