Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1395 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #99 |
| commutator | C2×C4 | C42×C13 |
| Frattini, cosocle | C2×#41 | C22×C13 |
| socle | C2×C13 | #342 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C4×C13 ◁ C2×C13×#41 ◁ G | |
162 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 162 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 — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,24,26,36)(17,28,29,21)(18,31)(19,23)(20,25)(22,32)(27,41,40,37)(30,35,43,42)(33,45)(34,39)(38,44);(14,16,18,27,19,26,32,40)(15,20,22,33,23,34,31,44)(17,25,30,38,29,39,43,45)(21,24,35,37,28,36,42,41)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 127);