Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#1764 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | D4 |
| commutator | C4 | C2×C42×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C22×C13 | C4×D4 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C22×C8×C13 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,17)(18,19,22,26,23,27,21,24)(20,25,29,32,30,33,28,31);(15,17)(19,24,27,26)(25,31,33,32);(18,20,22,29,23,30,21,28)(19,25,26,32,27,33,24,31)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 496);