Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22×C4 |
| direct product | C11×#1963 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C2×D4 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C22×C11 | C2×C22⋊C4 |
| prime radical | C11×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C2×C42×C11 ◁ G | |
320 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 320 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)(16,17,20,23,21,24,19,29)(18,25,28,30,26,31,27,22);(16,18)(17,22,24,30)(19,27)(20,28)(21,26)(23,25,29,31);(12,13,14,15)(18,26)(22,30)(25,31)(27,28)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 695);