Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12738 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#102 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C22×C19×#135 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C23×C4×C19 ◁ C23×C8×C19 ◁ G | |
926 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)(22,23)(24,25,27,31,29,34,41,33)(26,35,38,49,39,52,36,51)(28,42,40,32,44,48,53,50)(30,43,46,37,47,54,45,55);(24,26)(25,30)(27,36)(28,43)(29,39)(31,45)(32,49)(33,46)(34,47)(35,48)(37,53)(38,41)(40,55)(42,52)(44,54)(50,51);(21,23)(24,27,29,41)(25,31,34,33)(26,36,39,38)(28,40,44,53)(30,45,47,46)(32,48,50,42)(35,51,52,49)(37,43,55,54)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5726);