Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#1276 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C13 | D4 |
| commutator | C4 | C4×C8×C13 |
| Frattini, cosocle | C2×C42 | C22×C13 |
| socle | C22×C13 | #95 |
| prime radical | C23×C13 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C13 ◁ C2×C42×C13 ◁ G | |
208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,18)(17,20)(19,21)(22,23,24,26,25,27,28,29);(15,18,20,19)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 8);