Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#1328 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #99 |
| commutator | C23 | C2×C8×C13 |
| Frattini, cosocle | C2×M16 | C22×C13 |
| socle | C2×C13 | #324 |
| prime radical | C23×C13 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C4×C13 ◁ C2×C13×M16 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,22,18,23,28,36,19,21,27,33,29,34,40,43)(16,20,25,30,26,32,39,41,24,31,37,35,38,42,45,44);(14,16)(15,20,21,31)(17,25)(18,26)(19,24)(22,35,33,30)(23,32,34,42)(27,37)(28,39)(29,38)(36,44,43,41)(40,45)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 60);