Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#418 ⋊ C2 |
| Sylow | C53 ⋊ #2008 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16810532 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#418 | C2 |
| socle | C2×C53 | #392 |
| hypercenter | #418 | D53 |
| hypocenter | C53 | #2008 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #418 | |
380 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|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=eg595 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2008, C2×#20275770, #27411097, … |
| maximal normal subgroup of | D53×#2008, C2×#20275770, #27411097, … |
| maximal quotient of | #27284402, #27284416, #27284430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5565);