Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#564 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C53 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C53 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ G | |
172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(56,62)(57,65)(59,63)(60,67)(70,71,74,77)(72,75,80,78)(73,81,82,76)(79,84,85,83);(54,55,58,61)(56,63,62,59)(57,60,65,67)(64,69,68,66)(70,71)(72,78)(73,81)(74,77)(75,80)(76,82)(79,83)(84,85);(54,56,58,62)(55,59,61,63)(57,64,65,68)(60,66,67,69)(70,72)(71,75)(73,79)(74,80)(76,83)(77,78)(81,84)(82,85);(54,57)(55,60)(56,64)(58,65)(59,66)(61,67)(62,68)(63,69)(70,73,74,82)(71,76,77,81)(72,79,80,85)(75,83,78,84)1007 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 245);