Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C13 · Schur multiplier (H2) = C2×C42 |
| direct product | C13×#1726 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C13 | C22 |
| commutator | C2 | C43×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| socle, prime radical | C23×C13 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C13 ◁ G | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(22,23,25,28,26,29,32,35)(24,27,31,34,30,33,36,37);(14,15,16,17)(22,24)(23,27)(25,31)(26,30)(28,34)(29,33)(32,36)(35,37);(18,19,20,21)(24,30)(27,33)(31,36)(34,37)1040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 458);