Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C24 |
| direct product | C4×C13×#125 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C13 | C23 |
| commutator | C22 | C23×C4×C13 |
| Frattini, cosocle | C23 | C24×C13 |
| socle, prime radical | C23×C13 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C13 ◁ G | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33);(7,11)(9,12)(13,14,16,18)(15,17,19,20);(5,6,8,10)(7,9,11,12)(13,14)(15,17)(16,18)(19,20);(5,7)(6,9)(8,11)(10,12)(13,15,16,19)(14,17,18,20)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1037);