Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1381 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | D8 |
| commutator | C8 | C42×C13 |
| Frattini, cosocle | C22×C8 | C22×C13 |
| socle | C22×C13 | #107 |
| prime radical | C23×C13 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 2 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,17)(18,19,22,27,24,30,36,29)(20,31,33,45,35,46,47,40)(21,28,23,26,37,43,38,44)(25,34,39,48,42,49,41,32);(15,17)(18,20,23,34,22,33,37,48,24,35,38,49,36,47,21,32)(19,25,28,40,29,41,44,46,30,42,43,45,27,39,26,31)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 113);