Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#2082 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C23×C13 |
| prime radical | C23×C13 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C22×C4×C13 ◁ G | |
184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(14,15,19,24,20,25,18,23)(16,26,29,21,30,37,28,38)(17,31,33,39,34,40,35,22)(27,42,43,45,32,36,44,41);(14,16,20,30)(15,21,25,38)(17,32,34,27)(18,29,19,28)(22,36,39,42)(23,37,24,26)(31,45,40,41)(33,43,35,44);(14,17,18,33,20,34,19,35)(15,22,23,31,25,39,24,40)(16,27,28,43,30,32,29,44)(21,36,26,41,38,42,37,45)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 814);