Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C13 · Schur multiplier (H2) = C1 |
| direct product | C13×#1373 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | D8 |
| commutator | C8 | C2×C8×C13 |
| Frattini, cosocle | C4×C8 | C22×C13 |
| socle | C2×C13 | #335 |
| prime radical | C22×C13 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C2×C8×C13 ◁ C4×C8×C13 ◁ G | |
62 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)(14,15,18,23,19,25,17,33,20,26,28,22,34,24,35,40)(16,27,30,37,31,42,29,21,32,36,43,38,44,41,45,39);(14,16,28,43,19,31,35,45,20,32,18,30,34,44,17,29)(15,21,33,41,24,37,23,36,26,39,40,42,25,38,22,27)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 105);