Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of C22⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊(C7⋊C27) |
| unambiguous semidirect product | C7⋊(C22⋊C27) |
| derived, Hall | C22×C7 ⋊ C27 |
| Sylow | C7 ⋊ C22⋊C27 |
| Sylow | C22 ⋊ C7⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C9 | C22⋊ΓL(1,𝔽8) |
| commutator, hypocenter | C22×C7 | C27 |
| Fitting, cosocle | C22×C7×C9 | C3 |
| socle, prime radical | C22×C3×C7 | C9 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C9 | |
22 conjugacy classes of subgroups.
| transitive degree | μt(G) = 378 |
Polycyclic presentation:
3^3,2^2,7|a^3=b,b^3=c,d^a=de,e^a=d,f^a=f^2108 conjugacy classes — view the full character table.
| maximal subgroup of | D7⋊(C22⋊C27), C2×C22⋊(C7⋊C27) |
| maximal normal subgroup of | D7⋊(C22⋊C27), C2×C22⋊(C7⋊C27) |
| maximal quotient of | C7⋊(Q8⋊C27), C2×C22⋊(C7⋊C27) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 11);