Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of D21 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C32×Dic21 |
| unambiguous semidirect product | C3⋊(C32×Dic7) |
| derived | C3×C7 ⋊ C32×C4 |
| Hall | C33×C7 ⋊ C4 |
| Sylow | C33 ⋊ Dic7 |
| Sylow | C7 ⋊ C32×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D21 |
| commutator, hypocenter | C3×C7 | C32×C4 |
| Frattini | C2 | C32×D21 |
| Fitting, socle, prime radical | C2×C33×C7 | C2 |
| cosocle | C2×C3×C7 | C2×C32 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×C32 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 252 |
216 conjugacy classes — view the full character table.
| maximal subgroup of | C32×Dic3×D7, C32×D3×Dic7, C32×#4245, … |
| maximal normal subgroup of | C32×Dic3×D7, C32×D3×Dic7, C32×#4245, … |
| maximal quotient of | C32×C3⋊(C7⋊C8), C2×C32×Dic21 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 144);