Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#556 |
| 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 | |
212 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 1696 |
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)(56,58)(57,60)(59,61)(64,68)(66,69);(54,56,59,57)(55,60,61,58)(62,63,65,67)(64,66,68,69);(54,57,59,56)(55,60,61,58)(62,64)(63,66)(65,68)(67,69);(54,58,59,60)(55,56,61,57)1325 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 237);