Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#3293 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C13 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ G | |
444 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,21,26)(16,27,30,32)(17,29,22,36)(18,23,35,41)(19,25,20,24)(28,43,45,39)(31,44,38,33)(34,42,37,40);(14,16,21,30)(15,22,26,17)(18,33,35,44)(19,36,20,29)(23,39,41,43)(24,32,25,27)(28,37,45,34)(31,40,38,42);(14,17,21,22)(15,16,26,30)(18,31,35,38)(19,32,20,27)(23,28,41,45)(24,29,25,36)(33,42,44,40)(34,43,37,39);(14,18)(15,23)(16,28)(17,31)(19,37)(20,34)(21,35)(22,38)(24,42)(25,40)(26,41)(27,43)(29,44)(30,45)(32,39)(33,36)377 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2025);