Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13139 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #481 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
318 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)(21,22)(24,25)(26,34,37,50)(27,30)(28,40)(29,43)(31,36,45,52)(32,42)(33,35)(38,53,44,48)(39,54)(41,47,55,49)(46,51);(20,21,23,22)(25,30)(28,39)(31,45)(33,43)(36,52)(41,55)(42,51)(47,49);(20,22,23,21)(24,26,35,38,27,37,29,44)(25,31,43,47,30,45,33,49)(28,41,51,52,39,55,42,36)(32,48,54,50,46,53,40,34)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6127);