Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C8×C49 ⋊ C2 |
| Sylow | C49 ⋊ M16 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D49 |
| commutator | C2×C49 | C2×C4 |
| Frattini | C4×C7 | C2×D7 |
| Fitting | C8×C49 | C2 |
| socle | C2×C7 | C4×D7 |
| prime radical | C2×D49 | C4 |
| cosocle | C4×C49 | C22 |
| hypercenter | C8 | D49 |
| hypocenter | C49 | M16 |
| LCS | G ▷ C2×C49 ▷ C49 | |
| UCS | C1 ◁ C4 ◁ C8 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 392 |
Polycyclic presentation:
2^4,7^2|b^2=c,c^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 | C2×#11715985, #13095931, M16×D49, … |
| maximal normal subgroup of | C2×#11715985, #13095931, M16×D49, … |
| maximal quotient of | Dic49⋊C8, #13095856, (C2×D49)⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 5);