Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#12904 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#106 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
318 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)(26,34,36,46)(27,38)(28,31)(29,42)(30,35,43,48)(32,40)(33,41)(37,51,50,47)(39,52)(44,55,54,49)(45,53);(24,26,27,35,29,37,40,49,28,36,39,48,41,50,53,55)(25,30,32,34,33,44,38,47,31,43,45,46,42,54,52,51);(20,21,22,23)(25,31)(30,43)(32,45)(33,42)(34,46)(38,52)(44,54)(47,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5892);