Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C19 · Schur multiplier (H2) = C25×C4 |
| direct product | C19×#20422 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C4 | C22×C42×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×M16×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)(20,21)(22,23)(24,25,29,33,30,34,28,43)(26,35,39,45,37,49,38,31)(27,32,42,47,40,46,41,53)(36,48,51,54,52,55,50,44);(21,23)(24,26)(25,31,34,45)(27,36)(28,38)(29,39)(30,37)(32,44,46,54)(33,35,43,49)(40,52)(41,50)(42,51)(47,48,53,55);(24,27,29,42,30,40,28,41)(25,32,33,47,34,46,43,53)(26,36,39,51,37,52,38,50)(31,44,35,48,45,54,49,55);(26,37)(27,40)(31,45)(32,46)(35,49)(38,39)(41,42)(47,53)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13410);