Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#1391 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #99 |
| commutator | C23 | C42×C11 |
| Frattini, cosocle | C2×#41 | C22×C11 |
| socle | C2×C11 | #342 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C4×C11 ◁ C2×C11×#41 ◁ G | |
166 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(14,22)(15,26,27,19)(16,20)(17,21)(18,23,30,36)(24,34)(25,35)(28,39,40,32)(29,33)(31,37,41,43)(38,42);(12,14,16,24,17,25,29,38)(13,18,20,31,21,30,33,41)(15,23,28,37,27,36,40,43)(19,22,32,34,26,35,39,42)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 123);