Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#11857 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#102 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,28,30,25)(23,26)(24,27)(29,34,31,33)(32,35)(37,43,45,48)(38,39)(40,41)(44,46,51,47)(49,50);(20,22)(21,25)(23,31)(24,30)(26,34)(27,28)(29,32)(33,35)(36,37,42,45)(38,48,50,43)(39,51,49,44)(40,47,41,46);(20,23,24,32)(21,26,27,35)(22,29,30,31)(25,33,28,34)(36,38,41,39,42,50,40,49)(37,44,47,43,45,51,46,48)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4845);