Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22×C4 |
| direct product | C11×#1967 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C22×D4×C11 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C2×C42×C11 ◁ G | |
312 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 312 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) = 31 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,15)(16,17,20,23,19,22,26,28)(18,21,25,30,24,29,27,31);(16,18,20,27,19,24,26,25)(17,21,28,30,22,29,23,31);(12,13,14,15)(17,22)(21,29)(23,28)(30,31)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 699);