Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #2967 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | #125×C53 | C4 |
| hypercenter | #125 | #6157 |
| hypocenter | C53 | #2967 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #125 | |
362 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 148 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,c^2=f,d^2=e,f^2=g,b^a=be,d^a=dg,h^a=h^23,c^b=cg,h^b=h^52,h^c=h^30,h^d=h^52,h^f=h^52226 conjugacy classes — view the full character table.
| maximal subgroup of | #28305029, #28305031, #28305033, … |
| maximal normal subgroup of | #28305029, #28305031, #28305033, … |
| maximal quotient of | #28318464, #28318473, #28318529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21334);