Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×C49 ⋊ C2 |
| Sylow | C49 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D49 |
| commutator, cosocle | C2×C49 | C23 |
| Frattini, socle | C2×C7 | C22×D7 |
| Fitting | Q8×C49 | C2 |
| hypercenter | Q8 | D49 |
| hypocenter | C49 | #41 |
| LCS | G ▷ C2×C49 ▷ C49 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
60 conjugacy classes of subgroups.
| transitive degree | μt(G) = 392 |
Polycyclic presentation:
2^4,7^2|b^2=d,c^2=d,e^7=f,c^a=cd,e^a=e^6f^6,f^a=f^6,c^b=cd130 conjugacy classes — view the full character table.
| maximal subgroup of | #13095943, #13095945, #13095947, … |
| maximal normal subgroup of | #13095943, #13095945, #13095947, … |
| maximal quotient of | #13095920, #13095922, #13095923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 34);