Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#12642 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×D8 |
| commutator | C8 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C4×D8 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23)(24,25,28,35,31,36,30,46)(26,37,40,50,43,55,42,48)(27,33,44,52,45,53,29,34)(32,47,49,38,51,41,54,39);(21,23)(24,26,29,41,30,42,45,38,31,43,44,47,28,40,27,39)(25,32,33,50,35,49,52,55,36,51,53,48,46,54,34,37);(24,27,28,44,31,45,30,29)(25,33,35,52,36,53,46,34)(26,38,40,41,43,39,42,47)(32,48,49,37,51,50,54,55)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5630);