Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | M32×C53 ⋊ C4 |
| Sylow | C53 ⋊ #1379 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22⋊#6157 |
| commutator | C4×C53 | C4×C8 |
| Frattini | C2×C8 | C2×#6157 |
| Fitting | M32×C53 | C4 |
| socle | C2×C53 | #336 |
| hypercenter | M32 | #6157 |
| hypocenter | C53 | #1379 |
| LCS | G ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ M32 | |
136 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 54 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=g,d^2=f,f^2=g,b^a=bc,h^a=h^23,c^b=cg,d^b=dg,h^b=h^30,h^d=h^52,h^e=h^52316 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20289771, #28310267, #28310456, … |
| maximal normal subgroup of | C2×#20289771, #28310267, #28310456, … |
| maximal quotient of | #28298268, #28298278, #28298288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19566);