Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#2395 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C13 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C13 |
| socle | C23×C13 | C24 |
| prime radical | C22×D4×C13 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C13 ◁ G | |
560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19)(20,23)(21,22)(24,25)(26,27)(28,30)(29,31)(32,33);(14,15)(16,17)(19,22)(23,25)(26,28,29,32)(27,30,31,33);(15,17)(27,31)(30,33);(18,20,21,24)(19,23,22,25)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1127);