Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×C9×Dic7 |
| derived, Sylow | C7 ⋊ C3×C4×C9 |
| Hall | C3×C7×C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C9 | D7 |
| commutator, hypocenter | C7 | C3×C4×C9 |
| Frattini | C2×C3 | C32×D7 |
| Fitting | C2×C3×C7×C9 | C2 |
| socle, prime radical | C2×C32×C7 | C2×C3 |
| cosocle | C2×C3×C7 | C2×C32 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3×C9 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 756 |
270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×#92586, C3×D9×Dic7, C3×D63⋊C4, … |
| maximal normal subgroup of | C3×#92586, C3×D9×Dic7, C3×D63⋊C4, … |
| maximal quotient of | C3×C9×C7⋊C8, C2×C3×C9×Dic7 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 38);