Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C23×C4 |
| direct product | C13×#2381 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C24 |
| commutator | C22 | C23×C4×C13 |
| Frattini, cosocle | C23 | C24×C13 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ G | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17)(19,22)(20,24)(27,31)(28,34)(33,36)(35,40);(19,22)(23,25)(26,27,30,31)(28,32,34,38)(29,36,37,33)(35,41,40,39);(18,19,21,22)(20,23,24,25)(26,28)(27,32)(29,35)(30,34)(31,38)(33,39)(36,41)(37,40);(18,20)(19,23)(21,24)(22,25)(26,29,30,37)(27,33,31,36)(28,35,34,40)(32,39,38,41)494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1113);