Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12827 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1804 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
376 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)(22,33,37,52)(23,38)(24,29)(25,45)(26,32)(27,35,51,68)(28,40)(30,43)(31,44)(34,62,65,83)(36,50,46,57)(39,71)(41,72)(42,58)(47,53)(48,56)(49,78,81,69)(54,76)(55,75)(59,74)(60,73,79,64)(61,77,82,67)(63,80,66,70);(20,22,26,37)(21,27)(23,34,41,65)(24,35,44,68)(25,46,48,36)(28,49)(29,33)(30,57)(31,52)(32,51)(38,70)(39,63,42,66)(40,73,75,64)(43,77,74,67)(45,61)(47,78,76,69)(50,59)(53,79)(54,60)(55,81)(56,82)(58,83)(62,71)(72,80);(20,23,26,41)(21,28,32,55)(22,34,37,65)(24,42,44,39)(25,47,48,76)(27,49,51,81)(29,54,31,53)(30,58,59,71)(33,60,52,79)(35,66,68,63)(36,69,46,78)(38,56,72,45)(40,74,75,43)(50,62,57,83)(61,70,82,80)(64,77,73,67)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5815);