Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C11×C32 · Schur multiplier (H2) = C22 |
| direct product | C11×C22⋊C32 |
| unambiguous semidirect product | C22⋊(C11×C32) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C11×C16 | C22 |
| commutator | C2 | C2×C11×C32 |
| Frattini, cosocle | C2×C16 | C22×C11 |
| socle | C22×C11 | C2×C16 |
| prime radical | C23×C11 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11×C16 ◁ G | |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(16,17,18,22,19,23,26,32,20,24,27,33,29,35,38,42,21,25,28,34,30,36,39,43,31,37,40,44,41,45,46,47);(13,15)880 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 131);