Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of #1088 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×#1088 |
| Fitting, Hall | C7×H(3,𝔽3) ⋊ C4 |
| derived, Sylow | H(3,𝔽3) ⋊ C4×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C3×C7 | #157 |
| commutator, hypocenter | H(3,𝔽3) | C4×C7 |
| Frattini | C3 | C7×#157 |
| Fitting | C7×H(3,𝔽3) | C4 |
| prime radical | C7×#272 | C2 |
| cosocle | #272 | C2×C7 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3×C7 | |
34 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
98 conjugacy classes — view the full character table.
| maximal subgroup of | C7×#6249, C7⋊#6249, D7×#1088, … |
| maximal normal subgroup of | C7×#6249, C7⋊#6249, D7×#1088, … |
| maximal quotient of | C7×#6188, C2×C7×#1088 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 86);