Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of D72 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×D72 |
| unambiguous semidirect product | (C7×D7)⋊C23 |
| unambiguous semidirect product | (C7⋊D7)⋊C23 |
| Fitting, socle | C22×C72 ⋊ C22 |
| derived, Sylow | C72 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D72 |
| commutator, cosocle, hypocenter | C72 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C22×C72 | C22 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C22 | |
316 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | D7×(C2×D7)⋊C4, #13096359, #13096360, … |
| maximal normal subgroup of | D7×(C2×D7)⋊C4, #13096359, #13096360, … |
| maximal quotient of | C2×D7×Dic14, C2×#11716099, C2×#11716100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 169);