Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#2098 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C2×C13 | #399 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ G | |
210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 80 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)(14,15,18,23,19,24,20,34)(16,25,27,37,29,41,28,21)(17,30,32,40,33,22,35,39)(26,42,31,45,43,38,44,36);(14,16,20,27,19,29,18,28)(15,21,34,25,24,37,23,41)(17,31,35,43,33,44,32,26)(22,38,40,36,30,42,39,45);(14,17)(15,22)(16,26)(18,32)(19,33)(20,35)(21,36)(23,39)(24,30)(25,42)(27,31)(28,44)(29,43)(34,40)(37,45)(38,41)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 830);