Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊Dic49 |
| Sylow | C49 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D49 |
| commutator | C2×C49 | C2×C4 |
| Frattini, socle, prime radical | C22×C7 | C2×D7 |
| Fitting | C2×C4×C49 | C2 |
| cosocle | C22×C49 | C22 |
| hypercenter | C2×C4 | D49 |
| hypocenter | C49 | C4⋊C4 |
| LCS | G ▷ C2×C49 ▷ C49 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
39 conjugacy classes of subgroups.
| transitive degree | μt(G) = 784 |
Polycyclic presentation:
2^4,7^2|a^2=c,b^2=d,e^7=f,b^a=bd,e^a=e^6f^6,f^a=f^6202 conjugacy classes — view the full character table.
| maximal subgroup of | #13095847, #13095848, #13095857, … |
| maximal normal subgroup of | #13095847, #13095848, #13095857, … |
| maximal quotient of | C4⋊(C49⋊C8), #13095853, #13095857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 12);