Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11393 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #379 |
| prime radical | C24×C19 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(28,29,33,37,34,38,32,45)(30,39,41,35,42,50,46,49)(31,36,43,52,44,53,47,51)(40,55,58,59,57,54,56,48);(20,21,23,25)(22,26,27,24)(28,30)(29,35)(31,40)(32,46)(33,41)(34,42)(36,48)(37,50)(38,49)(39,45)(43,58)(44,57)(47,56)(51,54)(52,55)(53,59);(20,22,23,27)(21,24,25,26)(28,31)(29,36)(30,40)(32,47)(33,43)(34,44)(35,48)(37,52)(38,53)(39,54)(41,58)(42,57)(45,51)(46,56)(49,59)(50,55)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4381);