Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C53×#342 ⋊ C2 |
| Sylow | C53 ⋊ #1890 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133187 |
| commutator | C23×C53 | C22×C4 |
| Frattini | C24 | C22×D53 |
| Fitting | C53×#342 | C2 |
| socle | C22×C53 | #120 |
| hypercenter | #342 | D53 |
| hypocenter | C53 | #1890 |
| LCS | G ▷ C23×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #342 | |
386 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 83 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=dfg,b^a=bd,c^a=ce,e^a=eg,e^b=eg,h^b=h^52,d^c=dg,f^c=fg,h^c=h^52604 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#1890, C2×#20275127, #27385713, … |
| maximal normal subgroup of | D53×#1890, C2×#20275127, #27385713, … |
| maximal quotient of | #27278781, #27278791, #27279070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 4922);