Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C11 · Schur multiplier (H2) = C27 |
| direct product | C11×C22⋊#95 |
| unambiguous semidirect product | C22⋊(C11×#95) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C11 | C23 |
| commutator | C22 | C2×C42×C11 |
| Frattini, cosocle | C24 | C23×C11 |
| prime radical | C25×C11 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C11 ◁ G | |
1088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,15)(16,17,18,19)(21,23);(20,21)(22,23)(24,25,26,27);(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 165);