Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#1369 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #106 |
| commutator | C8 | C2×C8×C11 |
| Frattini, cosocle | C4×C8 | C22×C11 |
| socle | C2×C11 | #334 |
| prime radical | C22×C11 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C8×C11 ◁ C4×C8×C11 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,22,18,23,33,39,19,24,32,21,17,35,15,31)(14,25,27,36,29,40,26,37,30,38,34,43,28,42,41,20);(12,14,15,26,17,28,32,27,19,30,33,41,18,29,16,34)(13,20,21,25,23,36,31,40,24,37,22,38,35,43,39,42)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 101);