Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#2053 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C2×C13 | #394 |
| prime radical | C13×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ G | |
264 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,25)(17,30)(18,35)(19,23)(20,24)(21,27)(22,32)(26,36)(28,38)(29,37)(31,39)(33,40)(34,43)(41,44)(42,45);(14,16)(15,21)(17,31)(18,28)(19,27)(20,29)(22,39)(23,37)(24,38)(25,35)(26,33)(30,36)(32,42)(34,41)(40,44)(43,45);(14,17,18,32,20,34,19,33)(15,22,23,30,24,40,35,43)(16,26,27,41,29,42,28,31)(21,36,25,39,38,45,37,44)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 785);