Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C4×C19×#365 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #106 |
| commutator | C8 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×#106 |
| prime radical | C23×C19 | C2×SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(6,10,11,16)(7,13)(8,15)(12,19,18,17)(14,20);(5,6,7,10,8,12,13,17,9,11,14,16,15,18,20,19)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5657);