Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#3059 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C13 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C2×C13×#41 ◁ G | |
460 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 200 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,17)(18,19,23,27)(20,28,32,42)(21,25,33,39)(22,35,29,26)(24,36,38,31)(30,41,44,49)(34,46,43,40)(37,47,48,45);(18,20,29,36,23,32,22,31)(19,24,35,42,27,38,26,28)(21,30,43,47,33,44,34,45)(25,37,46,49,39,48,40,41);(15,17)(18,21)(19,25)(20,30)(22,34)(23,33)(24,37)(26,40)(27,39)(28,41)(29,43)(31,45)(32,44)(35,46)(36,47)(38,48)(42,49);(21,33)(25,39)(30,44)(34,43)(37,48)(40,46)(41,49)(45,47)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1791);