Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of 3−1+2⋊D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | 3−1+2⋊Dic7 |
| derived | C7×C9 ⋊ C3×C4 |
| Hall | C7×3−1+2 ⋊ C4 |
| Sylow | 3−1+2 ⋊ Dic7 |
| Sylow | C7 ⋊ 3−1+2⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | 3−1+2⋊D7 |
| commutator, hypocenter | C7×C9 | C3×C4 |
| Frattini | C2×C3 | C3×D21 |
| Fitting | C2×C7×3−1+2 | C2 |
| socle | C2×C3×C7 | C3×D3 |
| prime radical | C2×C32×C7 | D3 |
| cosocle | C2×C7×C9 | C2×C3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 ◁ C2 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 252 |
Polycyclic presentation:
2,3,2,3,7,3|a^2=c,d^3=f,d^a=d^2f^2,e^a=e^6,f^a=f^2,d^b=df^286 conjugacy classes — view the full character table.
| maximal subgroup of | #13094347, Dic7×AGL(1,ℤ/9), D7×3−1+2⋊C4, … |
| maximal normal subgroup of | #13094347, Dic7×AGL(1,ℤ/9), D7×3−1+2⋊C4, … |
| maximal quotient of | 3−1+2⋊(C7⋊C8), C2×3−1+2⋊Dic7 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 48);