Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#11026 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #390 |
| prime radical | C22×C19×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
1278 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)(26,34,37,30)(27,38,41,31)(28,33)(29,32)(35,45)(36,50,42,46)(39,47)(40,51,43,48)(44,49)(52,53)(54,55);(21,23)(24,26)(25,30)(27,39)(28,42)(29,37)(31,47)(32,34)(33,50)(35,40)(36,44)(38,55)(41,54)(43,53)(45,48)(46,49)(51,52);(24,27)(25,31)(26,35)(28,43)(29,41)(30,45)(32,38)(33,51)(34,52)(36,54)(37,53)(39,42)(40,44)(46,55)(47,50)(48,49)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4014);