Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #459 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
324 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)(20,21,23,22)(24,25)(26,34)(27,39)(28,44)(29,46)(30,35)(31,37)(32,41)(33,45)(36,50)(38,52)(40,54)(42,51)(43,55)(47,49)(48,53);(20,22,23,21)(24,26,28,37,30,38,45,29)(25,31,33,34,35,46,44,52)(27,36,40,53,43,41,42,47)(32,39,48,51,50,55,49,54);(24,27,28,40,30,43,45,42)(25,32,33,48,35,50,44,49)(26,36,37,53,38,41,29,47)(31,39,34,51,46,55,52,54)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6123);