Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#12839 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #427 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1808 |
| prime radical | C2×#41×C19 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C2×M16×C19 ◁ C2×C19×#131 ◁ G | |
362 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)(22,32)(23,38,40,28)(24,42)(25,45)(26,30)(27,35)(29,33)(31,43)(34,59,61,52)(36,65)(37,54)(39,69,71,72)(41,50,70,68)(44,73)(46,60,66,82)(47,64)(48,75)(49,57,67,79)(51,58)(53,56)(55,63)(62,77,81,80)(74,83,76,78);(20,22,33,56,25,36,44,64,26,37,58,75,43,63,24,35)(21,27,42,55,31,48,51,54,30,47,73,65,45,53,29,32)(23,34,57,78,41,62,71,82,40,61,79,83,70,81,39,60)(28,46,72,77,50,74,67,59,38,66,69,80,68,76,49,52);(20,23)(21,28)(22,34)(24,39)(25,41)(26,40)(27,46)(29,49)(30,38)(31,50)(32,52)(33,57)(35,60)(36,62)(37,61)(42,72)(43,70)(44,71)(45,68)(47,66)(48,74)(51,67)(53,76)(54,59)(55,77)(56,78)(58,79)(63,81)(64,82)(65,80)(69,73)(75,83)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5827);