Finite Group DatabasebetaFinite Group Databasebeta| order | 3392 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#533 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C53 |
| socle | C22×C53 | C24 |
| prime radical | C23×C53 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ G | |
222 conjugacy classes of subgroups.
| 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)(56,59)(57,58)(60,61)(62,63,65,67)(64,68,69,66);(55,58)(59,61);(62,64,65,69)(63,66,67,68);(54,56,57,60)(55,59,58,61)1484 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3392, 214);