Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #120⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#379⋊C3 |
| unambiguous semidirect product | #379⋊(C3×C7) |
| Fitting, Hall | C7×#379 ⋊ C3 |
| derived | C24 ⋊ C3×C4×C7 |
| Sylow | #379 ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | #120⋊C3 |
| commutator, hypocenter | C24 | C3×C4×C7 |
| Frattini | C23 | C2×C7×A4 |
| Fitting | C7×#379 | C3 |
| socle | C23×C7 | C2×A4 |
| prime radical | C2×C7×C22⋊A4 | C2 |
| cosocle | C25 | C2×C3×C7 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2×C7 | |
212 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,4,5,6,7)(8,12,18,13,10,16,11,17,14,9,15,19);(9,13)(16,19)140 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#379⋊C3, C7×AΣL(1,𝔽64), #72277810, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#379⋊C3, C7×AΣL(1,𝔽64), #72277810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4334);