Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C26 |
| direct product | C13×#2614 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C13 |
| prime radical | C22×D4×C13 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ G | |
782 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 782 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(19,21)(22,23,26,29)(24,30,32,27)(25,28,34,33)(31,36,37,35);(15,17)(19,21)(22,24,26,32)(23,27,29,30)(25,31,34,37)(28,35,33,36);(14,15)(16,17)(18,19)(20,21)(22,23)(24,30)(25,33)(26,29)(27,32)(28,34)(31,35)(36,37);(22,25)(23,28)(24,31)(26,34)(27,35)(29,33)(30,36)(32,37)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1346);