Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#3091 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C13 |
| socle | C22×C13 | C22×D4 |
| prime radical | C13×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C23×C4×C13 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,19)(17,18)(20,21)(22,23)(24,31)(25,29)(26,34)(27,30)(28,32)(33,36)(35,37);(15,18)(19,21);(14,16,17,20)(15,19,18,21)(22,24)(23,28)(25,32)(26,35)(27,33)(29,36)(30,37)(31,34);(22,25,27,26)(23,29,30,34)(24,32,33,35)(28,36,37,31)494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1823);