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×#563 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C53 |
| prime radical | C2×D4×C53 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ G | |
196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 77 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)(55,58)(56,60)(62,63,67,65)(64,70,72,68)(66,73,75,69)(71,76,74,77);(55,58)(59,61)(62,64,67,72)(63,68,65,70)(66,74,75,71)(69,77,73,76);(54,55)(56,59)(57,58)(60,61)(62,65,67,63)(64,68,72,70)(66,73,75,69)(71,76,74,77);(54,56)(55,59)(57,60)(58,61)(62,66)(63,69)(64,71)(65,73)(67,75)(68,76)(70,77)(72,74)1007 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 244);