Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#1639 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C13 |
| prime radical | D42×C13 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C42×C13 ◁ G | |
308 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,18)(17,20)(19,21)(22,23,26,29)(24,27,25,28);(15,18)(16,21)(19,20)(23,27)(24,25)(28,29);(23,28)(24,25)(27,29)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 371);