Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#1384 |
| 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 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
132 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)(16,26,28,40)(17,23)(18,33)(19,35)(20,25)(21,27,36,41)(22,31)(24,34)(29,44,42,38)(30,45,43,37)(32,39);(14,16,18,29,34,43,32,41,20,28,31,42,19,30,17,27)(15,21,23,37,35,44,22,26,25,36,39,45,24,38,33,40)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 116);