Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#13625 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #357 |
| commutator | C16 | C22×C4×C19 |
| Frattini, cosocle | C2×C16 | C23×C19 |
| socle | C2×C19 | C2×#357 |
| prime radical | C2×C19×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C22×C8×C19 ◁ C22×C16×C19 ◁ G | |
290 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)(20,21,23,29,27,32,26,50)(22,33,36,66,38,68,34,64)(24,41,39,76,45,63,44,81)(25,46,40,62,49,59,48,31)(28,51,53,35,55,71,52,73)(30,57,56,69,60,80,78,43)(37,74,70,58,75,83,47,67)(42,61,79,82,72,65,77,54);(20,22)(21,28)(23,34)(24,42)(25,47)(26,36)(27,38)(29,52)(30,58)(31,61)(32,55)(33,63)(35,69)(37,48)(39,77)(40,75)(41,68)(43,73)(44,79)(45,72)(46,54)(49,70)(50,53)(51,80)(56,74)(57,71)(59,82)(60,67)(62,65)(64,81)(66,76)(78,83);(20,23,27,26)(21,29,32,50)(22,34,38,36)(24,39,45,44)(25,40,49,48)(28,52,55,53)(30,56,60,78)(31,46,62,59)(33,64,68,66)(35,51,73,71)(37,47,75,70)(41,76,63,81)(42,77,72,79)(43,57,69,80)(54,65,82,61)(58,74,67,83)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6613);