Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1347 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #102 |
| commutator | C2×C8 | C2×C4×C13 |
| Frattini, cosocle | C22×C8 | C22×C13 |
| prime radical | C13×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
126 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 416 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,20,21,18)(17,19)(22,23)(24,27)(25,34)(26,30)(28,31)(29,32)(33,36)(35,37);(14,16)(15,18)(17,21)(19,20)(23,27)(24,31)(25,32)(28,29)(30,36)(33,37)(34,35)377 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 79);