Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#1381 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | D8 |
| commutator | C8 | C42×C11 |
| Frattini, cosocle | C22×C8 | C22×C11 |
| socle | C22×C11 | #107 |
| prime radical | C23×C11 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
128 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)(16,17,20,25,22,28,34,27)(18,29,31,43,33,44,45,38)(19,26,21,24,35,41,36,42)(23,32,37,46,40,47,39,30);(13,15)(16,18,21,32,20,31,35,46,22,33,36,47,34,45,19,30)(17,23,26,38,27,39,42,44,28,40,41,43,25,37,24,29)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 113);