Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | D3×D7×H(5,𝔽2) |
| Fitting | C3×C7×H(5,𝔽2) ⋊ C22 |
| Hall | C3×C7 ⋊ C22×H(5,𝔽2) |
| Sylow | C7 ⋊ C2×D3×H(5,𝔽2) |
| Sylow | C3 ⋊ C2×D7×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×D3×D7 |
| commutator, socle, cosocle | C2×C3×C7 | C26 |
| Fitting | C3×C7×H(5,𝔽2) | C22 |
| hypercenter | H(5,𝔽2) | D3×D7 |
| hypocenter | C3×C7 | C22×H(5,𝔽2) |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
13120 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
255 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×B(4,𝔽2), D3×D7×#453, D3×H(5,𝔽2)×AGL(1,𝔽7), … |
| maximal normal subgroup of | D3×D7×B(4,𝔽2), D3×D7×#453, D3×H(5,𝔽2)×AGL(1,𝔽7), … |
| maximal quotient of | D7×#10596675 |
Past the SmallGroups library coverage — no SmallGroups id.