Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#1521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| prime radical | C13×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C2×C42×C13 ◁ G | |
218 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 88 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)(15,20,22,34)(16,25)(17,30)(21,36,37,42)(23,35,24,31)(26,41,38,33)(28,44)(29,40)(32,43)(39,45);(14,15)(16,26)(17,31)(18,23)(19,24)(20,32)(21,29)(22,27)(25,37)(28,41)(30,34)(33,44)(35,43)(36,45)(38,40)(39,42);(14,16,27,40)(15,21,22,37)(17,28,43,44)(18,29,19,25)(20,33,34,41)(23,26,24,38)(30,45,32,39)(31,36,35,42)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 253);