Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of #5997 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C7 |
| direct product | C22×#5997 |
| Fitting, socle | C22×C72 ⋊ C4 |
| derived, Sylow | C72 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #5997 |
| commutator, hypocenter | C72 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C72 | C4 |
| prime radical | C22×C7⋊D7 | C2 |
| cosocle | C7⋊D7 | C23 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C22 | |
118 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | #13096190, #13096200, #13096219, … |
| maximal normal subgroup of | #13096190, #13096200, #13096219, … |
| maximal quotient of | C2×#11716091, C2×#11716092, #13096665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 168);