Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1333 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #102 |
| commutator | C8 | C2×C8×C13 |
| Frattini, cosocle | C4×C8 | C22×C13 |
| prime radical | C2×C13×D8 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C42×C13 ◁ C4×C8×C13 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,18,17,19,20,21)(22,23,25,29,26,31,36,30)(24,32,34,28,35,27,33,37);(23,27)(24,33)(25,36)(28,29)(30,37)(31,32)(34,35)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 65);