Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C53 ⋊ C2×#405 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×#6157 |
| commutator | C2×C53 | C22×C42 |
| Frattini | C22 | C23×#6157 |
| Fitting | #117×C53 | C4 |
| socle | C22×C53 | C23×C4 |
| hypercenter | #117 | #6157 |
| hypocenter | C53 | C2×#405 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ #117 | |
592 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=e,b^2=f,e^2=g,h^a=h^30,c^b=cg,h^d=h^52,h^e=h^52340 conjugacy classes — view the full character table.
| maximal subgroup of | #28306370, #28306394, #28306399, … |
| maximal normal subgroup of | #28306370, #28306394, #28306399, … |
| maximal quotient of | #28313747, #28313753, #28313777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21070);