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) = C22 |
| direct product | C11×#1392 |
| 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 | |
158 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 158 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)(12,13,16,21,18,23,17,22)(14,24,26,39,28,41,27,40)(15,29,30,38,31,20,32,36)(19,33,34,25,35,42,37,43);(12,14,17,27,18,28,16,26)(13,19,22,34,23,35,21,37)(15,25,30,42,31,43,32,33)(20,24,36,40,29,41,38,39)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 124);