Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C13 · Schur multiplier (H2) = C1 |
| direct product | C13×#1398 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | C42 |
| commutator | C4 | C4×C8×C13 |
| Frattini, cosocle | C2×C16 | C22×C13 |
| socle | C2×C13 | #346 |
| prime radical | C22×C13 | M32 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,21,17,23,26,36,18,24,27,34,28,38,40,44,19,22,25,35,29,37,39,45,30,20,31,32,41,33,42,43);(15,20,22,24)(16,25)(21,34,35,32)(23,33,37,38)(26,39)(27,31)(36,44,45,43)(40,42)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 130);