Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#13622 |
| 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 | |
294 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)(22,33,36,64)(23,29)(24,40)(25,45)(26,49)(27,32)(28,35,52,70)(30,42)(31,47)(34,62,50,66)(37,73,72,83)(38,74)(39,60)(41,59,71,63)(43,79)(44,61)(46,65,69,56)(48,57)(51,67,54,81)(53,80,82,75)(55,76)(58,68)(77,78);(20,22)(21,28)(23,34)(24,41)(25,46)(26,50)(27,36)(29,51)(30,56)(31,59)(32,52)(33,61)(35,68)(37,47)(38,75)(39,72)(40,64)(42,70)(43,80)(44,71)(45,53)(48,69)(49,54)(55,73)(57,82)(58,65)(60,63)(62,79)(66,74)(67,78)(76,81)(77,83);(20,23,27,26)(21,29,32,49)(22,34,36,50)(24,38,44,43)(25,39,48,47)(28,51,52,54)(30,55,58,77)(31,45,60,57)(33,62,64,66)(35,67,70,81)(37,46,72,69)(40,74,61,79)(41,75,71,80)(42,76,68,78)(53,63,82,59)(56,73,65,83)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6610);