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) = C2 |
| direct product | C13×#1401 |
| subgroup | G/H | |
|---|---|---|
| center | C13×C16 | D4 |
| commutator | C4 | C2×C13×C16 |
| Frattini, cosocle | C2×C16 | C22×C13 |
| socle | C2×C13 | C22⋊C16 |
| prime radical | C13×#41 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C13×C16 ◁ C2×C13×C16 ◁ G | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 2 generators on 77 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,24,19,25,37,50,20,26,38,51,40,53,65,73,21,27,39,52,41,54,66,74,17,23,34,47,35,48,62,71)(16,28,30,55,31,56,59,75,29,45,57,49,58,69,76,72,32,46,36,63,60,70,64,77,33,22,42,43,61,44,67,68);(14,16)(15,22)(17,33)(18,36)(19,31)(20,29)(21,32)(23,46)(24,49)(25,44)(26,28)(27,45)(30,39)(34,57)(35,61)(37,64)(38,42)(40,58)(41,60)(43,52)(47,55)(48,70)(50,72)(51,63)(53,56)(54,69)(59,66)(62,76)(65,67)(68,74)(71,75)(73,77)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 133);