Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C13×C16 · Schur multiplier (H2) = C2 |
| direct product | C13×#1397 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C13×C16 | C22 |
| commutator | C2 | C4×C13×C16 |
| Frattini, cosocle | C2×C16 | C22×C13 |
| socle, prime radical | C22×C13 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13×C16 ◁ G | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 2 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19,20,25,21,26,28,37,22,27,29,38,31,39,40,47,23,24,30,34,32,35,41,44,33,36,42,45,43,46,48,49);(14,15,16,17)(19,24)(25,34)(26,35)(27,36)(37,44)(38,45)(39,46)(47,49)1040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 129);