Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4×C7 · Schur multiplier (H2) = C27 |
| direct product | C22×C3×C7×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3×C7 | C22 |
| commutator | C2 | C23×C3×C4×C7 |
| Frattini, cosocle | C22 | C24×C3×C7 |
| prime radical | C25×C3×C7 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3×C7 ◁ G | |
1348 conjugacy classes of subgroups; only the 676 normal subgroups are shown.
This group has 1348 conjugacy classes of subgroups — too many to draw. Only its 676 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2)(13,14,15)(16,17,18,19,20,21,22);(3,4);(5,6,8,10)(7,9,11,12);(6,9)(10,12)840 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11221);