Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12942 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #447 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1862 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 83 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,26,33,28,34,27,49)(22,35,39,53,40,66,50,62)(23,41,44,58,45,78,51,77)(24,32,47,63,48,64,25,31)(29,46,54,74,56,38,55,37)(30,57,59,79,61,42,60,43)(36,67,70,52,72,81,71,82)(65,75,83,73,80,68,76,69);(20,22,26,39,28,40,27,50)(21,29,33,54,34,56,49,55)(23,36,44,70,45,72,51,71)(24,46,47,74,48,38,25,37)(30,52,59,81,61,82,60,67)(31,62,32,35,63,53,64,66)(41,76,58,65,78,83,77,80)(42,73,43,68,57,69,79,75);(20,23)(21,30)(22,36)(24,43)(25,42)(26,44)(27,51)(28,45)(29,52)(31,41)(32,58)(33,59)(34,61)(35,65)(37,73)(38,75)(39,70)(40,72)(46,68)(47,57)(48,79)(49,60)(50,71)(53,83)(54,81)(55,67)(56,82)(62,76)(63,78)(64,77)(66,80)(69,74)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5930);