Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#2078 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C13 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ G | |
198 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 198 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) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,19)(17,18)(20,21)(22,23,25,30,27,31,26,29)(24,28,33,36,34,35,32,37);(15,18)(19,21)(22,24)(23,28)(25,33)(26,32)(27,34)(29,37)(30,36)(31,35);(14,16,17,20)(15,19,18,21)(23,29)(24,32)(25,26)(28,35)(30,31)(33,34)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 810);