Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×C7⋊C8 |
| unambiguous semidirect product | (C2×C7)⋊(C7×C8) |
| derived | C7 ⋊ C2×C7×C8 |
| Sylow | C72 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C7 | D7 |
| commutator, hypocenter | C7 | C2×C7×C8 |
| Frattini | C4 | C2×C7×D7 |
| Fitting | C2×C4×C72 | C2 |
| socle, prime radical | C22×C72 | C4 |
| cosocle | C4×C7 | C22×C7 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C4×C7 | |
68 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
280 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic7×C7⋊C8, #13096045, #13096046, … |
| maximal normal subgroup of | Dic7×C7⋊C8, #13096045, #13096046, … |
| maximal quotient of | C4×C7×C7⋊C8, C7×C4⋊(C7⋊C8), C2×C7×C7⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 69);