Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C42 |
No aliases for this group.
| Fitting | C4×C8×C53 ⋊ C4 |
| Sylow | C53 ⋊ #1274 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×#6157 |
| commutator | C4×C53 | C4×C8 |
| Frattini | C42 | C2×#6157 |
| Fitting | C4×C8×C53 | C4 |
| socle | C22×C53 | #95 |
| hypercenter | C4×C8 | #6157 |
| hypocenter | C53 | #1274 |
| LCS | G ▷ C4×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 | |
168 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 56 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=d,b^2=e,c^2=f,d^2=g,b^a=bc,e^a=ef,d^b=df,h^b=h^30,h^e=h^52460 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20289538, #28300082, #28300085, … |
| maximal normal subgroup of | C2×#20289538, #28300082, #28300085, … |
| maximal quotient of | #28297998, #28298003, #28298005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19333);