Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#2095 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C13 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C42×C13 ◁ G | |
254 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19,21,23)(20,24,25,22)(26,27,29,32,31,28,30,33);(14,15)(16,17)(18,20,21,25)(19,22,23,24)(26,27)(28,31)(29,33)(30,32);(15,17)(26,28,31,27)(29,32,30,33)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 827);