Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#5948 |
| Fitting, Hall | C22×C7×#41 ⋊ C3 |
| Sylow | C22×#41 ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C7 | C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C2×C3×C7 |
| Frattini | C2 | C2×C7×C22⋊A4 |
| Fitting | C22×C7×#41 | C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C4×C7 | |
350 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7)(9,10,11)(12,13,16,19)(14,18,22,25,15,24,21,27,20,17,23,26);(12,20,16,22)(13,24,19,26)(14,23,21,15)(17,27,25,18);(8,11)(9,10)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11609);