Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33455 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C22×D8 |
| commutator | C8 | C23×C4×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C19×#117 ◁ C19×#405 ◁ G | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2432 |
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)(20,21,23,22)(25,30)(27,37)(28,39)(31,43)(32,47)(33,44)(35,50)(36,49)(38,41)(40,42)(45,54)(46,55)(48,53)(51,52);(24,25,29,33)(26,31,34,45)(27,38,40,47)(28,41,39,32)(30,42,44,37)(35,51,48,55)(36,52,49,46)(43,53,54,50);(20,22,23,21)(24,26)(25,31)(27,35)(28,36)(29,34)(30,43)(32,46)(33,45)(37,50)(38,51)(39,49)(40,48)(41,52)(42,53)(44,54)(47,55);(26,34)(31,45)(35,48)(36,49)(43,54)(46,52)(50,53)(51,55)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26443);