Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C2×C11×#365 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #106 |
| commutator | C8 | C22×C4×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C23×C11 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
152 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29);(4,8,9,14)(5,11)(6,13)(10,17,16,15)(12,18);(3,4,5,8,6,10,11,15,7,9,12,14,13,16,18,17)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 886);