Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#1351 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #99 |
| commutator | C22×C4 | C2×C4×C11 |
| Frattini, cosocle | C2×C22⋊C4 | C22×C11 |
| prime radical | C24×C11 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C23×C11 ◁ C2×C11×C22⋊C4 ◁ G | |
162 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,16,17,18)(20,21)(22,26,27,24)(23,25);(12,13)(14,18)(15,17)(16,19)(21,24,25,26)(22,27)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 83);