Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C2×C13×#436 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C13 | C23 |
| commutator | C22 | C23×C4×C13 |
| Frattini, cosocle | C2×C4 | C24×C13 |
| socle | C23×C13 | C22×C4 |
| prime radical | C22×D4×C13 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×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)(7,8,11,9,10,12,14,13);(4,6)(8,12)(9,13);(7,9,11,12,10,13,14,8)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1660);