Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3×C7 · Schur multiplier (H2) = C29 |
| direct product | C2×C3×C7×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C24 |
| commutator, Frattini, cosocle | C2 | C25×C3×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
1796 conjugacy classes of subgroups; only the 1508 normal subgroups are shown.
This group has 1796 conjugacy classes of subgroups — too many to draw. Only its 1508 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(11,12,13)(14,15,16,17,18,19,20);(3,4)(5,9)(6,7)(8,10);(4,7)(8,9);(3,4)(5,8)(6,7)(9,10);(3,5)(4,8)(6,10)(7,9)714 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11681);