Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#1859 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #120 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ G | |
302 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,14)(17,21)(18,19)(22,23);(12,14,15,13)(16,17)(18,23)(19,21)(20,22);(16,17)(18,21)(19,23)(20,22)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 591);