Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#12964 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C19 |
| Frattini, cosocle | C22×D4 | C23×C19 |
| prime radical | C2×C19×#127 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C23×C19 ◁ C22×D4×C19 ◁ G | |
672 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22)(24,25)(26,28)(27,30)(29,31);(20,22,23,21)(26,31);(25,28)(26,31)(29,30)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5952);