Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C7 · Schur multiplier (H2) = C27 |
| direct product | C2×C3×C7×#120 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3×C7 | C23 |
| commutator, Frattini, cosocle | C22 | C24×C3×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C7 ◁ G | |
1324 conjugacy classes of subgroups; only the 420 normal subgroups are shown.
This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 420 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 — 4 generators on 20 points:
(1,2)(11,12,13)(14,15,16,17,18,19,20);(3,4)(5,7)(6,8)(9,10);(4,7)(8,10);(4,8)(7,10)588 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11230);