Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#12597 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #117 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1746 |
| prime radical | C23×C19 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C19×#405 ◁ G | |
258 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)(20,21,24,29,25,31,41,53,26,30,42,52,43,54,68,78)(22,32,35,56,36,57,61,80,34,27,44,46,60,47,70,73)(23,28,38,49,40,50,66,76,39,51,65,77,67,71,82,83)(33,55,59,79,37,62,64,81,58,45,69,72,63,48,74,75);(20,22)(21,27,30,32)(23,37,39,63)(24,35)(25,36)(26,34)(28,48)(29,46,52,56)(31,47,54,57)(33,40,58,67)(38,64,65,74)(41,61)(42,44)(43,60)(45,71)(49,75)(50,55)(51,62)(53,73,78,80)(59,66,69,82)(68,70)(72,83)(76,79)(77,81);(20,23)(21,28)(22,33,34,58)(24,38)(25,40)(26,39)(27,45,32,55)(29,49)(30,51)(31,50)(35,59,44,69)(36,37,60,63)(41,66)(42,65)(43,67)(46,72,56,79)(47,48,57,62)(52,77)(53,76)(54,71)(61,64,70,74)(68,82)(73,75,80,81)(78,83)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5585);