Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | #108×C53 ⋊ C4 |
| Sylow | C53 ⋊ #1773 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×#6157 |
| commutator | C4×C53 | C2×C42 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | #108×C53 | C4 |
| socle | C2×C53 | C4×C4⋊C4 |
| hypercenter | #108 | #6157 |
| hypocenter | C53 | #1773 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #108 | |
224 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 88 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=e,b^2=f,c^2=de,d^2=g,b^a=bd,c^a=cg,h^a=h^23,d^b=dg,e^b=eg,h^b=h^23,h^c=h^30,h^e=h^52,h^f=h^52226 conjugacy classes — view the full character table.
| maximal subgroup of | #28310705, #28310709, #28310712, … |
| maximal normal subgroup of | #28310705, #28310709, #28310712, … |
| maximal quotient of | #28302069, #28302071, #28302073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20319);