Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of D21⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | D21⋊(C3⋊D3) |
| unambiguous semidirect product | (C3⋊(C3⋊D3))⋊D7 |
| Fitting, socle, derived, Hall | C33×C7 ⋊ C22 |
| Sylow | C33 ⋊ C2×D7 |
| Sylow | C7 ⋊ D3×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C33×C7 | C22 |
| LCS | G ▷ C33×C7 | |
| UCS | C1 | |
152 conjugacy classes of subgroups.
| transitive degree | μt(G) = 126 |
Polycyclic presentation:
2^2,3^3,7|c^a=c^2,d^a=d^2,f^a=f^6,c^b=c^2,d^b=d^2,e^b=e^263 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7×C3⋊D3, D3×(C3⋊D3)⋊D7, C2×D21⋊(C3⋊D3), … |
| maximal normal subgroup of | D3×D7×C3⋊D3, D3×(C3⋊D3)⋊D7, C2×D21⋊(C3⋊D3), … |
| maximal quotient of | #13094824, #13094825, D21⋊(C3⋊Dic3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 176);