Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C2×C4×C13×SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C13 | D4 |
| commutator | C4 | C23×C4×C13 |
| Frattini, cosocle | C2×C4 | C24×C13 |
| socle | C23×C13 | C2×D4 |
| prime radical | C22×D4×C13 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C13 ◁ C2×C42×C13 ◁ G | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27);(3,4,5,6);(7,8,9,10,11,12,13,14);(8,10)(9,13)(12,14)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1669);