Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | D4×C49 ⋊ C2 |
| Sylow | C49 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D49 |
| commutator, cosocle | C4×C49 | C22 |
| Frattini | C4×C7 | C2×D7 |
| Fitting | D4×C49 | C2 |
| socle | C2×C7 | C22⋊D7 |
| hypercenter | D4 | D49 |
| hypocenter | C49 | D8 |
| LCS | G ▷ C4×C49 ▷ C2×C49 ▷ C49 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 392 |
Polycyclic presentation:
2^4,7^2|c^2=d,e^7=f,b^a=bc,c^a=cd,e^a=e^6f^6,f^a=f^6,c^b=cd127 conjugacy classes — view the full character table.
| maximal subgroup of | D8×D49, #13095940, #13095943, … |
| maximal normal subgroup of | D8×D49, #13095940, #13095943, … |
| maximal quotient of | #13095847, #13095850, #13095866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 15);