Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of D72 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | D7×C7⋊C8 |
| unambiguous semidirect product | (C7×D7)⋊C8 |
| derived, Sylow | C72 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D72 |
| commutator, hypocenter | C72 | C2×C8 |
| Fitting, cosocle | C4×C72 | C22 |
| socle | C2×C72 | C2×C4 |
| prime radical | C2×C7×D7 | C4 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C4 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
100 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D72, D7×#1131, #13096242, … |
| maximal normal subgroup of | C8×D72, D7×#1131, #13096242, … |
| maximal quotient of | D7×C7⋊C16, #13096035, Dic7×C7⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 48);