Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C4 |
| direct product | C13×#1707 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C13 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C13 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C42×C13 ◁ G | |
180 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)(15,18)(16,20)(19,21)(22,23,26,29)(24,30,33,37)(25,34,31,28)(27,35,36,32);(22,24,31,35,26,33,25,32)(23,27,34,37,29,36,28,30);(14,15,16,18,17,19,20,21)494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 439);