Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#2090 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C13 |
| prime radical | C23×C13 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ G | |
176 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,21)(17,19)(18,20)(22,23,26,29,27,31,25,30)(24,28,32,37,34,35,33,36);(14,16,18,17)(15,19,20,21)(22,24)(23,28)(25,33)(26,32)(27,34)(29,37)(30,36)(31,35);(14,17,18,16)(15,19,20,21)(23,29)(24,32)(25,26)(28,35)(30,31)(33,34)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 822);