Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#2986 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C24 |
| commutator | C22 | C23×C4×C13 |
| Frattini, cosocle | C2×C4 | C24×C13 |
| socle | C22×C13 | C23×C4 |
| prime radical | C23×C13 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ G | |
330 conjugacy classes of subgroups; only the 256 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 256 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) = 832 |
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)(16,22)(17,20)(18,21)(19,24)(23,28)(25,27)(26,29)(30,31,33,36,34,37,40,43)(32,35,39,42,38,41,44,45);(14,16,18,24)(15,19,21,22)(17,25,26,23)(20,28,29,27)(30,32)(31,35)(33,39)(34,38)(36,42)(37,41)(40,44)(43,45);(14,17,18,26)(15,20,21,29)(16,23,24,25)(19,27,22,28)(32,38)(35,41)(39,44)(42,45);(15,21)(19,22)(20,29)(27,28)(32,38)(35,41)(39,44)(42,45)650 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1718);