Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#7187 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | #99 |
| commutator | C23 | C4×C8×C19 |
| Frattini, cosocle | C22×C22⋊C4 | C22×C19 |
| prime radical | C25×C19 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C24×C19 ◁ C22×C22⋊C4×C19 ◁ G | |
678 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 678 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 1216 |
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)(24,25,26,28,27,29,30,31)(32,33)(34,38,39,36)(35,37);(20,21,22,23)(33,36)(37,38)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 175);