Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of C3⋊AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(Dic7⋊C9) |
| unambiguous semidirect product | (C7⋊C9)⋊Dic3 |
| unambiguous semidirect product | Dic21⋊C9 |
| derived | C3×C7 ⋊ C4×C9 |
| Hall | C3×C7⋊C9 ⋊ C4 |
| Sylow | C7 ⋊ C9×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | C3⋊AGL(1,𝔽7) |
| commutator, hypocenter | C3×C7 | C4×C9 |
| Fitting, socle, prime radical, cosocle | C2×C32×C7 | C2×C3 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×C3 | |
44 conjugacy classes of subgroups.
| transitive degree | μt(G) = 252 |
Polycyclic presentation:
2,3,2,3^2,7|a^2=c,b^3=d,e^a=e^2,f^a=f^6,f^b=f^272 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×D7⋊C9, D3×Dic7⋊C9, #4245⋊C9, … |
| maximal normal subgroup of | Dic3×D7⋊C9, D3×Dic7⋊C9, #4245⋊C9, … |
| maximal quotient of | C3⋊((C7⋊C8)⋊C9), C2×C3⋊(Dic7⋊C9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 30);