Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Fitting | C53×B(4,𝔽2) ⋊ C2 |
| Sylow | C53 ⋊ #3027 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #120×D53 |
| commutator | C23×C53 | C24 |
| Frattini | C23 | C23×D53 |
| Fitting | C53×B(4,𝔽2) | C2 |
| socle | C2×C53 | C2×#120 |
| hypercenter | B(4,𝔽2) | D53 |
| hypocenter | C53 | #3027 |
| LCS | G ▷ C23×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ B(4,𝔽2) | |
772 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 772 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|d^2=g,b^a=be,c^a=cf,d^a=dg,f^b=fg,e^c=eg,h^d=h^52445 conjugacy classes — view the full character table.
| maximal subgroup of | #27538168, #27538170, #27538186, … |
| maximal normal subgroup of | #27538168, #27538170, #27538186, … |
| maximal quotient of | #27358059, #27358061, #27358066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13910);