Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#1813 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #117 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C2×C13 | #380 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C13×M16 ◁ G | |
252 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 116 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)(14,15,19,23,20,24,18,33)(16,25,28,36,29,40,27,21)(17,30,31,37,32,22,34,38)(26,41,42,45,43,39,44,35);(14,16,18,27,20,29,19,28)(15,21,23,25,24,36,33,40)(17,26,31,42,32,43,34,44)(22,35,37,39,30,45,38,41);(14,17)(15,22)(16,26)(18,31)(19,34)(20,32)(21,35)(23,37)(24,30)(25,39)(27,42)(28,44)(29,43)(33,38)(36,45)(40,41)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 545);