Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C3×C7×#332 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3×C7 | C22⋊C4 |
| commutator | C2×C4 | C2×C3×C4×C7 |
| Frattini, cosocle | C42 | C22×C3×C7 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ C3×C42×C7 ◁ G | |
128 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1344 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(12,15)(13,17)(16,18)(19,20,22,24,23,26,21,25);(11,12,14,16)(13,18,17,15)(20,24)(21,22)(25,26)399 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 772);