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) = C1 |
| direct product | C13×#1422 |
| subgroup | G/H | |
|---|---|---|
| center | C13×C16 | D4 |
| commutator | C4 | C2×C13×C16 |
| Frattini, cosocle | C2×C16 | C22×C13 |
| socle | C2×C13 | C4⋊C16 |
| prime radical | C22×C13 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C13×C16 ◁ C2×C13×C16 ◁ G | |
58 conjugacy classes of subgroups.
| 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)(14,15,18,24,19,25,33,40,20,23,34,38,30,39,43,45,21,26,35,41,28,22,36,37,17,31,32,44,16,27,29,42);(14,16,17,28,21,30,20,19)(15,22,23,27,26,25,31,39)(18,29,32,36,35,43,34,33)(24,37,38,42,41,40,44,45)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 154);