Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C53×#533 ⋊ C2 |
| Sylow | C53 ⋊ Q8×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D53 |
| commutator, socle | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C53×#533 | C2 |
| hypercenter | #533 | D53 |
| hypocenter | C53 | Q8×#41 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ #533 | |
1068 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,d^b=df,h^d=h^52778 conjugacy classes — view the full character table.
| maximal subgroup of | #27276132, #27276136, #27350968, … |
| maximal normal subgroup of | #27276132, #27276136, #27350968, … |
| maximal quotient of | #27543403, #27543406, #27543514, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 18088);