Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#1157 |
| Fitting | C2×C4×C72 ⋊ C2 |
| Sylow | C72 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | C2×D7 |
| commutator, cosocle | C2×C7 | C23×C7 |
| Frattini | C2 | C22×C7×D7 |
| Fitting | C2×C4×C72 | C2 |
| socle | C2×C72 | C23 |
| hypercenter | C2×C4×C7 | D7 |
| hypocenter | C7 | C7×#41 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4×C7 ◁ C2×C4×C7 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
238 conjugacy classes — view the full character table.
| maximal subgroup of | #13096060, #13096061, C7×#6379, … |
| maximal normal subgroup of | #13096060, #13096061, C7×#6379, … |
| maximal quotient of | C4×C7×Dic14, C7×#6433, C7×#6435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 139);