Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C42 |
No aliases for this group.
| Sylow | C53 ⋊ #1747 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C42×D53 |
| commutator | C4×C53 | C2×C42 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×H(3,ℤ/4) | C2 |
| socle | C2×C53 | #376 |
| hypercenter | H(3,ℤ/4) | D53 |
| hypocenter | C53 | #1747 |
| LCS | G ▷ C4×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ H(3,ℤ/4) | |
200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|a^2=e,b^2=f,c^2=d,d^2=g,b^a=bd,f^a=fg,e^b=eg,h^c=h^52616 conjugacy classes — view the full character table.
| maximal subgroup of | #27294625, #27294627, #27294632, … |
| maximal normal subgroup of | #27294625, #27294627, #27294632, … |
| maximal quotient of | #27274530, #27274532, #27274534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 4264);