Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1388 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #102 |
| commutator | C8 | C42×C13 |
| Frattini, cosocle | C22×C8 | C22×C13 |
| socle | C2×C13 | #340 |
| prime radical | C23×C13 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
124 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,24,21,26,20,37)(16,27,29,39,32,41,31,45)(17,33,35,42,36,25,19,23)(22,30,38,34,40,44,43,28);(14,16,19,30,20,31,36,28,21,32,35,44,18,29,17,34)(15,22,25,39,24,38,23,41,26,40,33,45,37,43,42,27)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 120);