Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#344 ⋊ C2 |
| Sylow | C53 ⋊ #2009 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16810533 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#344 | C2 |
| socle | C2×C53 | #392 |
| hypercenter | #344 | D53 |
| hypocenter | C53 | #2009 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #344 | |
348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,h^a=h^52,c^b=cf,d^b=dg,h^b=h^52,d^c=dg,e^c=eg,h^c=h^52595 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2009, C2×#20275778, #27411097, … |
| maximal normal subgroup of | D53×#2009, C2×#20275778, #27411097, … |
| maximal quotient of | #27284410, #27284424, #27284438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5573);