Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| direct product | D7×#997 |
| Fitting, derived | C7×H(5,𝔽2) ⋊ C2×C3 |
| Hall | D7×H(5,𝔽2) ⋊ C3 |
| Sylow | C7 ⋊ C2×#997 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×C22⋊A4 |
| commutator, Fitting, cosocle, hypocenter | C7×H(5,𝔽2) | C2×C3 |
| socle | C2×C7 | C2×C22⋊A4 |
| LCS | G ▷ C7×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
386 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 56 |
55 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#997, D7×#5935, D7×#5934, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#997, D7×#5935, D7×#5934, … |
| maximal quotient of | C2×D7×#997, C3×D7×#997, D21×#997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11495);