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) = C2 |
| direct product | C13×#1350 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #102 |
| commutator | C2×C8 | C2×C4×C13 |
| Frattini, cosocle | C22×C8 | C22×C13 |
| prime radical | C23×C13 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
104 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 2 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,20,21,18)(17,19)(22,23,26,31,28,33,27,32)(24,34,36,47,38,51,37,50)(25,39,41,30,43,48,42,49)(29,40,44,35,46,52,45,53);(14,16)(15,18)(17,21)(19,20)(22,24,28,38)(23,29,33,46)(25,40,43,52)(26,37,27,36)(30,47,49,50)(31,45,32,44)(34,39,51,48)(35,41,53,42)377 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 82);