Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#2080 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C2×C13 | #397 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ G | |
190 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 190 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)(16,24,27,37)(17,21)(18,22)(19,23)(20,26,34,31)(25,40,28,39)(29,35)(30,36)(32,38)(33,42,44,41)(43,45);(14,16)(15,20)(17,28)(18,31)(19,27)(21,33)(22,37)(23,34)(24,38)(25,30)(26,32)(29,41)(35,40)(36,44)(39,45)(42,43);(14,17,18,29,19,30,32,43)(15,21,22,35,23,36,38,45)(16,25,26,41,27,28,31,42)(20,33,24,39,34,44,37,40)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 812);