Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×B(4,𝔽2)⋊C3 |
| unambiguous semidirect product | B(4,𝔽2)⋊(C3×C7) |
| Fitting, Hall | C7×B(4,𝔽2) ⋊ C3 |
| derived | H(5,𝔽2) ⋊ C2×C3×C7 |
| Sylow | B(4,𝔽2) ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C7 | #120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C2×C3×C7 |
| Frattini | C23 | C2×C7×A4 |
| Fitting | C7×B(4,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2×C7 | |
116 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3,4,5,6,7)(9,14,13,12,10,15);(8,13)(9,10)(11,15)(12,14)112 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×B(4,𝔽2)⋊C3, D7×B(4,𝔽2)⋊C3, C7×#73026, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×B(4,𝔽2)⋊C3, D7×B(4,𝔽2)⋊C3, C7×#73026, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4344);