Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | H(5,𝔽2)×D21 |
| Fitting | C3×C7×H(5,𝔽2) ⋊ C2 |
| Hall | C3×C7 ⋊ C2×H(5,𝔽2) |
| Sylow | C7 ⋊ D3×H(5,𝔽2) |
| Sylow | C3 ⋊ D7×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×D21 |
| commutator, socle, cosocle | C2×C3×C7 | C25 |
| Fitting | C3×C7×H(5,𝔽2) | C2 |
| hypercenter | H(5,𝔽2) | D21 |
| hypocenter | C3×C7 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
1796 conjugacy classes of subgroups; only the 581 normal subgroups are shown.
This group has 1796 conjugacy classes of subgroups — too many to draw. Only its 581 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
204 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×H(5,𝔽2), D21×B(4,𝔽2), #72244285, … |
| maximal normal subgroup of | D3×D7×H(5,𝔽2), D21×B(4,𝔽2), #72244285, … |
| maximal quotient of | #72246737 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11671);