Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#22457 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C2×C19×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ G | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 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)(22,32)(23,37,40,28)(24,29)(25,44)(26,31)(27,35)(30,47)(33,60,63,58)(34,59)(36,56)(38,67,71,51)(39,52,74,68)(41,75)(42,57)(43,55)(45,50)(46,54)(48,62,66,72)(49,65)(53,64)(61,77,81,80)(69,76,82,83)(70,78,73,79);(20,22,26,36)(21,27,31,50)(23,33,40,63)(24,41,43,65)(25,45,47,35)(28,48,37,66)(29,53,55,59)(30,56,44,32)(34,46,64,42)(38,69,71,82)(39,72,74,62)(49,57,75,54)(51,77,67,80)(52,60,68,58)(61,73,81,70)(76,79,83,78);(20,23)(21,28)(22,33)(24,38)(25,39)(26,40)(27,48)(29,51)(30,52)(31,37)(32,58)(34,61)(35,62)(36,63)(41,69)(42,70)(43,71)(44,68)(45,72)(46,73)(47,74)(49,76)(50,66)(53,77)(54,78)(55,67)(56,60)(57,79)(59,80)(64,81)(65,82)(75,83);(20,24,26,43)(21,29,31,55)(22,34,36,64)(23,38,40,71)(25,46,47,42)(27,49,50,75)(28,51,37,67)(30,57,44,54)(32,59,56,53)(33,61,63,81)(35,65,45,41)(39,73,74,70)(48,76,66,83)(52,79,68,78)(58,80,60,77)(62,82,72,69)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 15445);