Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13173 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #466 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1942 |
| prime radical | C23×C19 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
262 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)(20,21,24,32,27,33,26,30)(22,34,38,51,39,56,36,28)(23,40,42,72,45,73,44,71)(25,47,46,58,48,57,41,31)(29,52,54,75,55,74,53,43)(35,61,65,80,66,81,63,70)(37,67,68,79,69,50,62,60)(49,64,59,77,76,83,78,82);(20,22,24,38,27,39,26,36)(21,28,32,34,33,51,30,56)(23,35,42,65,45,66,44,63)(25,37,46,68,48,69,41,62)(29,49,54,59,55,76,53,78)(31,50,47,60,58,67,57,79)(40,70,72,61,73,80,71,81)(43,64,52,77,75,83,74,82);(20,23,41,74,26,44,48,75,27,45,46,52,24,42,25,43)(21,29,47,72,32,54,58,73,33,55,57,71,30,53,31,40)(22,35,62,82,36,63,69,83,39,66,68,77,38,65,37,64)(28,49,60,61,34,59,67,80,51,76,79,81,56,78,50,70)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6161);