Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7×C16 · Schur multiplier (H2) = C22 |
| direct product | C3×C7×C22⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C7×C8 | C22 |
| commutator | C2 | C2×C3×C7×C16 |
| Frattini, cosocle | C2×C8 | C22×C3×C7 |
| socle | C22×C3×C7 | C2×C8 |
| prime radical | C23×C3×C7 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7×C8 ◁ G | |
132 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12)(13,14)(15,16,17,20,18,21,23,26,19,22,24,27,25,28,29,30);(12,14)840 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 788);