Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#2231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C23×C13 |
| prime radical | C13×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C4×C8×C13 ◁ G | |
120 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 60 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)(14,15,20,25)(16,26,29,24)(17,30)(18,34,27,23)(19,21,36,38)(22,39)(28,42)(31,44)(32,43)(33,37)(35,40)(41,45);(14,16,27,36,20,29,18,19)(15,21,34,24,25,38,23,26)(17,28,32,45,33,39,31,35)(22,37,41,43,42,30,40,44);(14,17)(15,22)(16,28)(18,31)(19,35)(20,33)(21,37)(23,40)(24,43)(25,42)(26,44)(27,32)(29,39)(30,38)(34,41)(36,45)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 963);