Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C13×C16 · Schur multiplier (H2) = C22 |
| direct product | C13×D4⋊C16 |
| unambiguous semidirect product | D4⋊(C13×C16) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | D4 |
| commutator | C4 | C2×C13×C16 |
| Frattini, cosocle | C4×C8 | C22×C13 |
| socle | C22×C13 | C22⋊C8 |
| prime radical | C2×D4×C13 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C4×C8×C13 ◁ G | |
110 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,18)(17,20)(19,21)(22,23,24,27,25,28,30,33,26,29,31,34,32,35,36,37);(15,18)(16,21)(19,20)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 61);