Finite Group DatabasebetaFinite Group Databasebeta| order | 756 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C27 · Schur multiplier (H2) = C2 |
| direct product | C2×D7×C27 |
| Fitting | C2×C7×C27 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C27 |
| Hall | C7×C27 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C27 | D7 |
| commutator, hypocenter | C7 | C22×C27 |
| Frattini | C9 | C2×C3×D7 |
| Fitting | C2×C7×C27 | C2 |
| socle | C2×C3×C7 | C2×C9 |
| prime radical | C2×C3×D7 | C9 |
| cosocle | C7×C9 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C27 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 378 |
270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D7×Dic27, #13094120, #13094121, … |
| maximal normal subgroup of | D7×Dic27, #13094120, #13094121, … |
| maximal quotient of | C27×Dic14, C4×D7×C27, C27×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(756, 12);