Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#1902 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #386 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C22×C4×C13 ◁ G | |
304 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 108 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)(17,29,32,39)(18,33)(19,23)(20,24)(21,27)(22,31,38,30)(26,41,43,35)(28,36)(34,37)(40,42,45,44);(14,16,18,27,20,28,19,34)(15,21,23,25,24,37,33,36)(17,26,31,44,32,43,30,42)(22,35,39,40,38,41,29,45);(14,17)(15,22)(16,26)(18,31)(19,30)(20,32)(21,35)(23,39)(24,38)(25,40)(27,44)(28,43)(29,33)(34,42)(36,45)(37,41)377 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 634);