Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12841 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #466 |
| 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 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
358 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)(23,40,43,54)(24,44)(25,31)(26,48)(27,32)(28,36)(29,42,53,47)(30,34)(35,62,65,81)(37,58)(38,69)(39,59)(41,52,72,79)(45,55)(46,74)(49,76,77,83)(50,66)(51,67)(56,64,80,68)(57,60)(61,71,75,73)(63,70,82,78);(20,22,34,60,26,38,46,67,27,39,45,66,25,37,24,36)(21,28,44,58,31,50,55,59,32,51,74,69,48,57,30,33)(23,35,61,76,42,64,72,82,43,65,75,83,47,68,41,63)(29,49,73,81,54,78,79,80,53,77,71,62,40,70,52,56);(20,23)(21,29)(22,35)(24,41)(25,47)(26,42)(27,43)(28,49)(30,52)(31,54)(32,53)(33,56)(34,61)(36,63)(37,68)(38,64)(39,65)(40,48)(44,73)(45,75)(46,72)(50,78)(51,77)(55,79)(57,70)(58,81)(59,80)(60,76)(62,69)(66,83)(67,82)(71,74)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5829);