Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | D3×#6553 |
| Fitting | C3×C7×#41 ⋊ C22 |
| Hall | C3×C7 ⋊ C2×H(5,𝔽2) |
| Sylow | C7 ⋊ D3×H(5,𝔽2) |
| Sylow | C3 ⋊ C2×#6553 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D3×D7 |
| commutator, socle, cosocle | C2×C3×C7 | C25 |
| Fitting | C3×C7×#41 | C22 |
| hypercenter | #41 | D3×D7 |
| hypocenter | C3×C7 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ #41 | |
1796 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
141 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×H(5,𝔽2), D3×#89970, D3×#89594, … |
| maximal normal subgroup of | D3×D7×H(5,𝔽2), D3×#89970, D3×#89594, … |
| maximal quotient of | #72246690, #72246700 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11556);