Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#1391 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #99 |
| commutator | C23 | C42×C13 |
| Frattini, cosocle | C2×#41 | C22×C13 |
| socle | C2×C13 | #342 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C4×C13 ◁ C2×C13×#41 ◁ G | |
166 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(16,24)(17,28,29,21)(18,22)(19,23)(20,25,32,38)(26,36)(27,37)(30,41,42,34)(31,35)(33,39,43,45)(40,44);(14,16,18,26,19,27,31,40)(15,20,22,33,23,32,35,43)(17,25,30,39,29,38,42,45)(21,24,34,36,28,37,41,44)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 123);