Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#509 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C23×C53 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×C8×C53 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 848 |
A minimal faithful permutation representation — 3 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)(56,59)(57,66)(58,60)(61,63)(62,64)(65,67)(68,69);(55,59)(56,63)(57,64)(60,67)(61,62)(65,69)(66,68);(55,60)(59,67)(61,68)(62,66)848 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 190);