Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#2129 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | C2×#99 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C2×D4 | C23×C13 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C13×#41 ◁ G | |
238 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 84 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)(16,24,27,39)(17,21)(18,31,33,22)(19,23)(20,26)(25,40,28,42)(29,44,45,37)(30,38)(32,35)(34,41)(36,43);(14,16,19,27)(15,20,23,35)(17,28,30,25)(18,32,33,26)(21,36,38,34)(22,39,31,24)(29,41,45,43)(37,40,44,42);(14,17,19,30)(15,21,23,38)(16,25,27,28)(18,29,33,45)(20,34,35,36)(22,37,31,44)(24,40,39,42)(26,41,32,43)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 861);