Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C27 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Dic7⋊C27 |
| unambiguous semidirect product | (C7⋊C27)⋊C4 |
| derived, Sylow | C7 ⋊ C4×C27 |
| Hall | C7⋊C27 ⋊ C4 |
| Hall | Dic7 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C9 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C4×C27 |
| Fitting, cosocle | C2×C7×C9 | C2×C3 |
| socle, prime radical | C2×C3×C7 | C2×C9 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C9 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 756 |
Polycyclic presentation:
2,3,2,3^2,7|a^2=c,b^3=d,d^3=e,f^a=f^6,f^b=f^4126 conjugacy classes — view the full character table.
| maximal subgroup of | Dic14⋊C27, C4×D7⋊C27, C2×Dic7⋊C27, … |
| maximal normal subgroup of | Dic14⋊C27, C4×D7⋊C27, C2×Dic7⋊C27, … |
| maximal quotient of | (C7⋊C8)⋊C27, C2×Dic7⋊C27 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 1);