Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#23362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #2380 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C19×#3153 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ G | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 4 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)(22,26,32,42)(23,34,37,44)(24,30)(25,29)(27,38,45,47)(28,36)(31,43)(33,41,39,46)(35,40)(48,51)(49,50);(20,22)(21,26)(23,35)(24,32)(25,39)(27,43)(28,41)(29,46)(30,42)(31,47)(33,36)(34,40)(37,49)(38,48)(44,50)(45,51);(20,23,36,47,24,37,25,38)(21,27,29,34,30,45,28,44)(22,31,39,35,32,48,33,49)(26,40,41,43,42,50,46,51);(21,28,30,29)(22,32)(26,41,42,46)(27,44,45,34)(31,48)(33,39)(35,49)(40,43,50,51)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 16350);