Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25×C4 |
| direct product | C19×#21248 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×#121 |
| prime radical | C22×C19×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25,29,35,30,34,28,33)(26,31,37,45,38,47,43,46)(27,39,41,49,42,48,40,32)(36,50,51,44,52,54,53,55);(20,21,22,23)(24,26,29,43,30,38,28,37)(25,31,33,45,34,47,35,46)(27,36,41,53,42,52,40,51)(32,44,48,54,49,55,39,50);(24,27,28,40,30,42,29,41)(25,32,35,39,34,49,33,48)(26,36,37,51,38,52,43,53)(31,44,46,50,47,55,45,54);(24,28,30,29)(25,33,34,35)(26,37,38,43)(27,40,42,41)(31,45,47,46)(32,48,49,39)(36,51,52,53)(44,54,55,50)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14236);