Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1298 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C22⋊C4 |
| commutator | C2×C4 | C42×C13 |
| Frattini, cosocle | C23×C4 | C22×C13 |
| socle | C22×C13 | #95 |
| prime radical | C24×C13 | Q8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C23×C4×C13 ◁ G | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,20)(17,19)(18,21)(22,23,25,30,27,31,26,29)(24,32,33,28,35,36,34,37);(14,16)(15,18,19,20)(17,21)(22,24,26,34,27,35,25,33)(23,28,30,36,31,37,29,32)494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 30);