Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#13019 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C22×D8×C19 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
496 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)(21,23)(24,25,29,33,45,54,28,46,31,37,27,34,30,36,47,35)(26,38,41,52,48,49,44,32,43,51,39,55,42,50,40,53);(20,21,22,23)(24,26)(25,32)(27,44)(28,39)(29,40)(30,48)(31,43)(33,49)(34,50)(35,51)(36,55)(37,53)(38,46)(41,47)(42,45)(52,54);(25,33)(27,28)(29,47)(30,45)(32,49)(34,37)(35,54)(36,46)(38,55)(39,44)(40,41)(42,48)(50,53)(51,52)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6007);