Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C19 · Schur multiplier (H2) = C27 |
| direct product | C2×C19×#1729 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C19 | C23 |
| commutator | C22 | C22×C42×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C24×C19 | C22×C4 |
| prime radical | C25×C19 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C19 ◁ G | |
2350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(3,4)(5,6)(11,12,14,17,15,18,21,24)(13,16,20,23,19,22,25,26);(4,6)(7,8,9,10)(11,13)(12,16)(14,20)(15,19)(17,23)(18,22)(21,25)(24,26);(13,19)(16,22)(20,25)(23,26)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13322);