Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C25 |
| direct product | C2×C19×#1316 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | #99 |
| commutator | C23 | C22×C8×C19 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C19 |
| prime radical | C26×C19 | C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C24×C19 ◁ C22×C22⋊C4×C19 ◁ G | |
1886 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 1886 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(3,4,5,9,6,10,12,16)(7,8,11,14,13,15,17,18);(4,8)(10,15)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4411);