Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×#123×C53 ⋊ C2 |
| Sylow | C53 ⋊ #2022 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133185 |
| commutator | C24×C53 | C23 |
| Frattini | C24 | C22×D53 |
| Fitting | C2×#123×C53 | C2 |
| socle | C22×C53 | #120 |
| hypercenter | C2×#123 | D53 |
| hypocenter | C53 | #2022 |
| LCS | G ▷ C24×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C2×#123 | |
430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|b^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^c=dg,h^c=h^52754 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2022, C2×#20275843, #27411333, … |
| maximal normal subgroup of | D53×#2022, C2×#20275843, #27411333, … |
| maximal quotient of | #27284313, #27284316, #27284320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5638);