Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C22×C42 |
| direct product | C53×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C53 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ G | |
260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 4 generators on 69 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)(54,55,58,61)(56,63,64,60)(57,65,67,69)(59,62,68,66);(54,56,58,64)(55,60,61,63)(57,66,67,62)(59,65,68,69);(54,57)(55,56)(58,67)(59,60)(61,64)(62,65)(63,68)(66,69);(54,56)(55,57)(58,64)(59,65)(60,62)(61,67)(63,66)(68,69)1007 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 242);