Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#336 ⋊ C2 |
| Sylow | C53 ⋊ #1851 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D53 |
| commutator, socle | C23×C53 | C22×C4 |
| Fitting | C53×#336 | C2 |
| hypercenter | #336 | D53 |
| hypocenter | C53 | #1851 |
| LCS | G ▷ C23×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ #336 | |
324 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 105 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=f,b^2=f,f^2=g,b^a=bd,c^a=ce,h^a=h^52,c^b=cg,h^c=h^521084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D53×#1851, C2×#20274899, #27384756, … |
| maximal normal subgroup of | D53×#1851, C2×#20274899, #27384756, … |
| maximal quotient of | #27277938, #27277943, #27277949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 4694);