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) = C22 |
| direct product | C13×#1392 |
| 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 | |
158 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 158 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)(14,15,18,23,20,25,19,24)(16,26,28,41,30,43,29,42)(17,31,32,40,33,22,34,38)(21,35,36,27,37,44,39,45);(14,16,19,29,20,30,18,28)(15,21,24,36,25,37,23,39)(17,27,32,44,33,45,34,35)(22,26,38,42,31,43,40,41)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 124);