Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #1914 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16811571 |
| commutator | C23×C53 | C22×C4 |
| Frattini | C23 | C22×#6157 |
| Fitting | #120×C53 | C4 |
| socle | C2×C53 | #386 |
| hypercenter | #120 | #6157 |
| hypocenter | C53 | #1914 |
| LCS | G ▷ C23×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #120 | |
340 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 66 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=f,b^2=df,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^a=h^23,d^b=dg,e^b=eg,f^b=fg,h^b=h^30,d^c=dg,f^c=fg,h^f=h^52211 conjugacy classes — view the full character table.
| maximal subgroup of | #28311070, #28311071, #28311072, … |
| maximal normal subgroup of | #28311070, #28311071, #28311072, … |
| maximal quotient of | #28304372, #28304373, #28304374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20561);