Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×#334 ⋊ C2 |
| Sylow | C53 ⋊ #1594 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D4×D53 |
| commutator | C2×C4×C53 | C22×C4 |
| Frattini | C42 | C22×D53 |
| Fitting | C53×#334 | C2 |
| socle | C22×C53 | C4×D4 |
| hypercenter | #334 | D53 |
| hypocenter | C53 | #1594 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C42 ◁ #334 | |
176 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 63 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,c^2=ef,d^2=f,e^2=g,b^a=bd,c^a=cfg,d^a=df,c^b=cg,d^b=df,h^b=h^52,h^c=h^52760 conjugacy classes — view the full character table.
| maximal subgroup of | #27296047, #27296049, #27296061, … |
| maximal normal subgroup of | #27296047, #27296049, #27296061, … |
| maximal quotient of | #27274923, #27275354, #27278042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 3423);