Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#12807 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #410 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| socle | C2×C19 | #1799 |
| prime radical | C19×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C2×C8×C19 ◁ C2×C19×#131 ◁ G | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
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)(22,31)(23,35)(24,29,38,27)(25,30)(26,32)(28,37)(33,45,49,44)(34,46)(36,42,50,43)(39,47,40,51)(41,48);(20,22,23,32,25,34,37,48)(21,26,28,31,30,41,35,46)(24,33,36,47,38,49,50,51)(27,39,42,44,29,40,43,45);(21,27)(24,38)(26,39)(28,42)(29,30)(31,44)(33,49)(35,43)(36,50)(40,41)(45,46)(47,51)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5795);