Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C19 · Schur multiplier (H2) = C27 |
| direct product | C19×#20360 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C22×C22⋊C4 |
| commutator | C22 | C22×C42×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C2×C19 | C22×#95 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×#41×C19 ◁ G | |
1996 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)(22,30,32,26)(23,27)(24,29)(25,28)(31,42,44,37)(33,43,46,38)(34,39)(35,40)(36,41)(45,50,51,48)(47,49);(20,22,24,33,25,32,36,46)(21,26,29,38,28,30,41,43)(23,31,35,45,34,44,47,51)(27,37,40,48,39,42,49,50);(20,23)(21,27)(22,31)(24,35)(25,34)(26,37)(28,39)(29,40)(30,42)(32,44)(33,45)(36,47)(38,48)(41,49)(43,50)(46,51);(23,34)(27,39)(31,44)(35,47)(37,42)(40,49)(45,51)(48,50)1558 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13348);