Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#1365 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C13 | #102 |
| commutator | C2×C8 | C2×C4×C13 |
| Frattini, cosocle | C4×C8 | C22×C13 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C42×C13 ◁ C4×C8×C13 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,18)(16,20)(19,21)(22,23,25,28,27,31,24,32,26,30,33,36,34,37,35,29);(14,15,17,19)(16,21,20,18)(23,28)(24,33)(25,35)(27,34)(29,31)(30,36)(32,37)377 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 97);