Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C22×C19×#368 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C19 | D8 |
| commutator | C8 | C23×C4×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C23×C19 | #107 |
| prime radical | C24×C19 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C19 ◁ C23×C4×C19 ◁ C23×C8×C19 ◁ G | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(3,4);(5,6,9,14,11,15,10,25)(7,16,19,31,21,32,20,35)(8,22,23,29,24,36,17,13)(12,26,27,34,28,33,30,18);(5,7,17,33,10,20,24,34,11,21,23,26,9,19,8,18)(6,12,22,31,14,27,29,32,15,28,36,35,25,30,13,16)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26400);