Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#12996 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
512 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)(21,23)(24,25,27,33,28,34,43,48,30,36,44,47,29,35,41,31)(26,37,38,54,45,32,49,50,40,53,42,46,39,51,55,52);(20,21,22,23)(25,31)(27,41)(28,29)(32,46)(33,35)(34,47)(36,48)(37,52)(38,55)(39,45)(42,49)(43,44)(50,53)(51,54);(24,26)(25,32)(27,42)(28,45)(29,39)(30,40)(31,46)(33,52)(34,53)(35,37)(36,51)(38,44)(41,49)(43,55)(47,50)(48,54)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5984);