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×#13207 |
| 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 | |
486 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)(30,38)(31,43)(32,35)(33,47)(34,41)(36,46)(37,44)(39,51)(40,53)(42,59)(45,48)(49,56)(50,55)(52,58)(54,57);(25,27)(28,30,39,55,33,42,31,41,32,40,46,58,44,57,45,56)(29,34,48,59,37,50,36,38,35,49,43,54,47,52,51,53);(20,21,22,23)(29,35)(30,40)(36,51)(37,47)(41,56)(42,57)(43,48)(55,58)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6195);