Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1328 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #99 |
| commutator | C23 | C2×C8×C11 |
| Frattini, cosocle | C2×M16 | C22×C11 |
| socle | C2×C11 | #324 |
| prime radical | C23×C11 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C4×C11 ◁ C2×C11×M16 ◁ G | |
90 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)(12,13,15,20,16,21,26,34,17,19,25,31,27,32,38,41)(14,18,23,28,24,30,37,39,22,29,35,33,36,40,43,42);(12,14)(13,18,19,29)(15,23)(16,24)(17,22)(20,33,31,28)(21,30,32,40)(25,35)(26,37)(27,36)(34,42,41,39)(38,43)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 60);