Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#1722 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C13 |
| prime radical | C13×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C42×C13 ◁ G | |
198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
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)(16,19)(20,21)(22,23,26,29)(24,30,31,37)(25,33,34,28)(27,35,36,32);(22,24,25,32,26,31,34,35)(23,27,28,30,29,36,33,37);(14,15,19,21,17,20,16,18)(24,31)(27,36)(30,37)(32,35)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 454);