Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #1565 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×C4⋊#6157 |
| commutator | C4×C53 | C22×C8 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | #97×C53 | C4 |
| socle | C22×C53 | C2×C4⋊C4 |
| hypercenter | #97 | #6157 |
| hypocenter | C53 | #1565 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #97 | |
180 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 78 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=d,d^2=f,e^2=g,b^a=bd,d^a=df,h^a=h^30,c^b=cf,h^c=h^52,h^e=h^52304 conjugacy classes — view the full character table.
| maximal subgroup of | D53⋊#1565, C2×#20290267, #28316088, … |
| maximal normal subgroup of | D53⋊#1565, C2×#20290267, #28316088, … |
| maximal quotient of | #28300173, #28302094, #28302095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20062);