Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1385 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #106 |
| commutator | C8 | C42×C13 |
| Frattini, cosocle | C22×C8 | C22×C13 |
| socle | C2×C13 | #340 |
| prime radical | C23×C13 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
124 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)(14,15,18,24,21,26,20,36)(16,27,29,42,31,43,34,38)(17,25,19,35,33,41,32,23)(22,30,37,28,40,44,39,45);(14,16,19,30,18,34,33,45,21,31,32,44,20,29,17,28)(15,22,25,38,36,37,23,27,26,40,41,42,24,39,35,43)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 117);