Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C23 |
| direct product | C2×C7×#997 |
| Fitting, Hall | C2×C7×H(5,𝔽2) ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C2×C3×C7 |
| Sylow | C2×H(5,𝔽2) ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22×C7 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C2×C3×C7 |
| Frattini | C2 | C2×C7×C22⋊A4 |
| Fitting | C2×C7×H(5,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C22×C7 | |
386 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
154 conjugacy classes — view the full character table.
| maximal subgroup of | C2×ΓL(1,𝔽8)×#997, C2×D7×#997, C2×C7×#5935, … |
| maximal normal subgroup of | C2×ΓL(1,𝔽8)×#997, C2×D7×#997, C2×C7×#5935, … |
| maximal quotient of | D3×C7×#997 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11610);