Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#2110 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | C42 |
| commutator | C4 | C2×C42×C13 |
| Frattini, cosocle | C2×C8 | C23×C13 |
| socle | C2×C13 | C2×#97 |
| prime radical | C23×C13 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ G | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,21,18,23,29,37,19,22,30,36,31,38,44,45)(16,24,26,40,27,33,42,35,25,39,28,43,41,20,32,34);(14,16)(15,20,22,33)(17,28)(18,27)(19,25)(21,35,36,34)(23,24,38,39)(26,30)(29,32)(31,41)(37,43,45,40)(42,44);(16,25)(20,33)(24,39)(26,28)(27,41)(32,42)(34,35)(40,43)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 842);