Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#3084 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C13 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ G | |
376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,19)(16,20,21,18)(22,23,27,31)(24,28,33,37)(25,30,26,35)(29,34,36,32);(14,16,17,21)(15,18,19,20)(22,24)(23,28)(25,34)(26,32)(27,33)(29,35)(30,36)(31,37);(22,24)(23,29)(25,32)(26,34)(27,33)(28,35)(30,37)(31,36);(22,25,27,26)(23,30,31,35)(24,32,33,34)(28,29,37,36)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1816);