Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#2644 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C13 |
| prime radical | C22×D4×C13 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ G | |
420 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,18)(19,20)(22,23)(24,27)(25,28)(26,29)(30,31,34,36)(32,37,33,35);(14,15,17,18)(16,19,21,20)(22,24)(23,25)(26,28)(27,29)(30,32,34,33)(31,35,36,37);(14,15)(16,20)(17,18)(19,21)(22,24)(23,27)(25,29)(26,28)(30,31)(32,35)(33,37)(34,36);(14,16,17,21)(15,19,18,20)(22,25,26,27)(23,28,29,24)(30,33,34,32)(31,35,36,37)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1376);