Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1615 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C13 |
| prime radical | C13×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ G | |
208 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 64 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)(14,15,21,31)(16,25,36,45)(17,19,24,20)(18,30,37,41)(22,38,44,27)(23,34,33,35)(26,42,39,32)(28,40,29,43);(14,16)(15,22)(17,26)(18,32)(19,29)(20,28)(21,36)(23,40)(24,39)(25,30)(27,35)(31,44)(33,43)(34,38)(37,42)(41,45);(14,17,20,15)(16,26,28,22)(18,33,35,41)(19,31,21,24)(23,34,30,37)(25,42,40,38)(27,45,32,43)(29,44,36,39)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 347);