Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C11×C16 · Schur multiplier (H2) = C22 |
| direct product | C11×D4⋊C16 |
| unambiguous semidirect product | D4⋊(C11×C16) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C11 | D4 |
| commutator | C4 | C2×C11×C16 |
| Frattini, cosocle | C4×C8 | C22×C11 |
| socle | C22×C11 | C22⋊C8 |
| prime radical | C2×D4×C11 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C11 ◁ C4×C8×C11 ◁ G | |
110 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 110 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,16)(15,18)(17,19)(20,21,22,25,23,26,28,31,24,27,29,32,30,33,34,35);(13,16)(14,19)(17,18)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 61);