Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13240 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C19 | C2×D8 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 4864 |
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)(21,23)(24,25,26,27)(28,29)(30,41)(31,46)(32,49)(33,51)(34,52)(35,40)(36,43)(37,47)(38,42)(39,50)(44,54)(45,59)(48,56)(53,57)(55,58);(20,21,22,23)(25,27)(28,30,42,57,34,45,32,43,35,31,47,58,50,48,33,44)(29,36,51,59,39,53,37,41,40,54,49,56,52,55,38,46);(28,31,47,57,34,48,33,43,35,30,42,58,50,45,32,44)(29,36,49,56,39,53,38,46,40,54,51,59,52,55,37,41)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6228);