Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C53×#533 ⋊ C2 |
| Sylow | C53 ⋊ #2031 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #13133206 |
| commutator | C22×C4×C53 | C23 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#533 | C2 |
| socle | C2×C53 | #393 |
| hypercenter | #533 | D53 |
| hypocenter | C53 | #2031 |
| LCS | G ▷ C22×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #533 | |
244 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 67 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=d,b^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^c=dg,e^c=eg,f^c=fg,h^c=h^52754 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2031, C2×#20275884, #27411520, … |
| maximal normal subgroup of | D53×#2031, C2×#20275884, #27411520, … |
| maximal quotient of | #27287177, #27287182, #27287187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5679);