Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#1159 |
| Fitting | C72×D4 ⋊ C2 |
| Sylow | C72 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C22×D7 |
| commutator, cosocle | C2×C7 | C23×C7 |
| Frattini | C2 | C22×C7×D7 |
| Fitting | C72×D4 | C2 |
| socle | C2×C72 | C23 |
| hypercenter | C7×D4 | D7 |
| hypocenter | C7 | C7×#41 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C7 ◁ C7×D4 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
175 conjugacy classes — view the full character table.
| maximal subgroup of | #13096363, #13096367, #13096371, … |
| maximal normal subgroup of | #13096363, #13096367, #13096371, … |
| maximal quotient of | C7×#6440, C7×#6441, C7×#6448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 141);