Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #1650 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133187 |
| commutator | C42×C53 | C23 |
| Frattini | C42 | C22×D53 |
| Fitting | C53×D4⋊C8 | C2 |
| socle | C22×C53 | #120 |
| hypercenter | D4⋊C8 | D53 |
| hypocenter | C53 | #1650 |
| LCS | G ▷ C42×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ D4⋊C8 | |
200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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=g,b^2=g,c^2=fg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^52,d^b=df,e^b=eg,h^b=h^52,d^c=dg,e^c=eg754 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#1650, C2×#20273973, #27359928, … |
| maximal normal subgroup of | D53×#1650, C2×#20273973, #27359928, … |
| maximal quotient of | #27273040, #27273054, #27273068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 3768);