Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of #87656 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C7 |
| direct product | C2×#87656 |
| Fitting, socle | C2×C72 ⋊ C8 |
| derived, Sylow | C72 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #87656 |
| commutator, hypocenter | C72 | C2×C8 |
| Frattini | C1 | G |
| Fitting, socle | C2×C72 | C8 |
| prime radical | C2×C7⋊D7 | C4 |
| cosocle | #5997 | C22 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C2 | |
32 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
28 conjugacy classes — view the full character table.
| maximal subgroup of | (#5997⋊C2)⋊C4, #13096620, #87658⋊C4, … |
| maximal normal subgroup of | (#5997⋊C2)⋊C4, #13096620, #87658⋊C4, … |
| maximal quotient of | #13096627, #13096628, C4×#87656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 164);