Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13197 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #121 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1954 |
| prime radical | C2×C19×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ G | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,32,26,31,42,47,27,33,41,29,24,30,23,40)(22,34,38,50,36,46,35,49,39,48,43,51,37,28,44,45);(20,22,23,35,24,37,41,38,27,39,42,44,26,36,25,43)(21,28,40,50,30,48,29,45,33,46,47,51,31,34,32,49);(20,23,24,41,27,42,26,25)(21,29,31,40,33,32,30,47)(22,35,37,38,39,44,36,43)(28,45,34,50,46,49,48,51)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6185);